Aditya,
   thanks for the information adding  \setmathfont[modern]   It fixed the 
problem for the default bullet symbols.  But, it did not fix the case where I'm 
trying to use the $\diamond$ symbol as the bullet, and compile time errors 
occur.  I will attach the log file.  Hopefully you can identify what is missing.

  I apologize for my lack of knowledge about fonts details and handling at this 
poing, but am learning.  Here is the minimal exampple:

--------------------------------
\usemodule[simplefonts]
\setmainfont[ipaexm]
\mainlanguage[ja]

\setmathfont[modern]

\definesymbol[5][$\diamond$]

\starttext
  \startitemize[5,columns,unpacked,two,broad]
       \item 此の行にbulletを打つ。
       \item One more column for good measure
       \item 此れは何処に置かれるでしょうか
   \stopitemize
\stoptext

----------------------------------------------------------------

(bullet-problem.tex

ConTeXt  ver: 2012.05.30 11:26 MKIV  fmt: 2013.3.17  int: english/english

system          > cont-new.mkiv loaded
(/usr/local/texlive/2012/texmf-dist/tex/context/base/cont-new.mkiv
system          > beware: some patches loaded from cont-new.mkiv
)
system          > bullet-problem.top loaded

system          > options > start used options

used options    > % runtime options files (command line driven)
used options    > \unprotect
used options    > % feedback and basic job control
used options    > % handy for special styles
used options    > \startluacode
used options    > document = document or { }
used options    > document.arguments={
used options    >  ["pdf"]=true,
used options    > }
used options    > document.files={
used options    >  "bullet-problem.tex",
used options    > }
used options    > \stopluacode
used options    > % process info
used options    > \setupsystem[inputfile=bullet-problem.tex]
used options    > \setupsystem[\c!n=1,\c!m=1]
used options    > % modes
used options    > % options (not that important)
used options    > \startsetups *runtime:options
used options    > \stopsetups
used options    > % styles and modules
used options    > \startsetups *runtime:modules
used options    > \stopsetups
used options    > % done
used options    > \protect \endinput

system          > options > stop used options

(bullet-problem.top)
fonts           > latin modern fonts are not preloaded
languages       > language en is active
resolvers       > modules > loaded: 'simplefonts'
(/usr/local/texlive/2012/texmf-dist/tex/context/third/simplefonts/t-simplefonts.mkiv
loading         > ConTeXt User Module / Simplefonts
<+ 
/usr/local/texlive/2012/texmf-dist/tex/context/third/simplefonts/t-simplefonts.lua>)
fonts           > typescripts > unknown: library 'loc'
{/usr/local/texlive/2012/texmf-dist/fonts/map/dvips/lm/lm-math.map}{/usr/local/texlive/2012/texmf-dist/fonts/map/dvips/lm/lm-rm.map}{/usr/local/texlive/2012/texmf-dist/fonts/map/pdftex/context/mkiv-base.map}
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:1-12pt-mm-mr--3]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:1-12pt-mm-mr--2]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:1-12pt-mm-mr--1]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:1-12pt-mm-mb--3]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:1-12pt-mm-mb--2]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:1-12pt-mm-mb--1]
fonts           > math: unset for global bodyfont simplefonts:1 at 12pt
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:2-12pt-mm-mr--3]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:2-12pt-mm-mr--2]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:2-12pt-mm-mr--1]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:2-12pt-mm-mb--3]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:2-12pt-mm-mb--2]
fonts           > defining > forced type otf of latinmodernmath-regular not 
found
fonts           > defining > font with asked name 'latinmodernmath-regular' is 
not found using lookup 'file'
fonts           > defining > unknown font latinmodernmath-regular, loading 
aborted
fonts           > defining > unable to define latinmodernmath-regular.otf as 
[simplefonts:2-12pt-mm-mb--1]
fonts           > math: unset for global bodyfont simplefonts:2 at 12pt
! Math error: parameter \Umathquad\displaystyle is not set.

system          > tex > error on line 13 in file bullet-problem.tex: Math 
error: parameter \Umathquad\displaystyle is not set ...

 3     \setmainfont[ipaexm]
 4     
 5     \mainlanguage[ja]
 6     
 7     \setmathfont[modern]
 8     
 9     \definesymbol[5][$\diamond$]
10     
11     \starttext
12       \startitemize[5,columns,unpacked,two,broad]
13 >>         \item 此の行にbulletを打つ。
14            \item One more column for good measure
15            \item 此れは何処に置かれるでしょうか
16        \stopitemize
17     \stoptext
18     


\47>:5 ->$\diamond $
                    
\symb_place_indeed ...name \??symbol #1\endcsname 
                                                  \relax \endgroup 
\symb_place_retry ...name \symb_place_indeed {:#1}
                                                  \else #1\fi 
\symb_place_normal_c ...lace_retry \currentsymbol 
                                                  \fi 
\symb_place_normal_b ...else \symb_place_normal_c 
                                                  \fi 
<to be read again> 
                   \else 
...
l.13        \item 此
                     の行にbulletを打つ。
? R
OK, entering \nonstopmode...
Missing character: There is no ⋄ (U+22C4) in font ipaexm!
! Math error: parameter \Umathquad\displaystyle is not set.

system          > tex > error on line 13 in file bullet-problem.tex: Math 
error: parameter \Umathquad\displaystyle is not set ...

 3     \setmainfont[ipaexm]
 4     
 5     \mainlanguage[ja]
 6     
 7     \setmathfont[modern]
 8     
 9     \definesymbol[5][$\diamond$]
10     
11     \starttext
12       \startitemize[5,columns,unpacked,two,broad]
13 >>         \item 此の行にbulletを打つ。
14            \item One more column for good measure
15            \item 此れは何処に置かれるでしょうか
16        \stopitemize
17     \stoptext
18     


\47>:5 ->$\diamond $
                    
\symb_place_indeed ...name \??symbol #1\endcsname 
                                                  \relax \endgroup 
\symb_place_retry ...name \symb_place_indeed {:#1}
                                                  \else #1\fi 
\symb_place_normal_c ...lace_retry \currentsymbol 
                                                  \fi 
\symb_place_normal_b ...else \symb_place_normal_c 
                                                  \fi 
<to be read again> 
                   \else 
...
l.13        \item 此
                     の行にbulletを打つ。
Sorry, but I can't typeset math unless various parameters have
been set. This is normally done by loading special math fonts
into the math family slots. Your font set is lacking at least
the parameter mentioned earlier.

! Math error: parameter \Umathquad\displaystyle is not set.

system          > tex > error on line 14 in file bullet-problem.tex: Math 
error: parameter \Umathquad\displaystyle is not set ...

 4     
 5     \mainlanguage[ja]
 6     
 7     \setmathfont[modern]
 8     
 9     \definesymbol[5][$\diamond$]
10     
11     \starttext
12       \startitemize[5,columns,unpacked,two,broad]
13            \item 此の行にbulletを打つ。
14 >>         \item One more column for good measure
15            \item 此れは何処に置かれるでしょうか
16        \stopitemize
17     \stoptext
18     


\47>:5 ->$\diamond $
                    
\symb_place_indeed ...name \??symbol #1\endcsname 
                                                  \relax \endgroup 
\symb_place_retry ...name \symb_place_indeed {:#1}
                                                  \else #1\fi 
\symb_place_normal_c ...lace_retry \currentsymbol 
                                                  \fi 
\symb_place_normal_b ...else \symb_place_normal_c 
                                                  \fi 
<to be read again> 
                   \else 
...
l.14        \item O
                   ne more column for good measure
Sorry, but I can't typeset math unless various parameters have
been set. This is normally done by loading special math fonts
into the math family slots. Your font set is lacking at least
the parameter mentioned earlier.

Missing character: There is no ⋄ (U+22C4) in font ipaexm!
! Math error: parameter \Umathquad\displaystyle is not set.

system          > tex > error on line 14 in file bullet-problem.tex: Math 
error: parameter \Umathquad\displaystyle is not set ...

 4     
 5     \mainlanguage[ja]
 6     
 7     \setmathfont[modern]
 8     
 9     \definesymbol[5][$\diamond$]
10     
11     \starttext
12       \startitemize[5,columns,unpacked,two,broad]
13            \item 此の行にbulletを打つ。
14 >>         \item One more column for good measure
15            \item 此れは何処に置かれるでしょうか
16        \stopitemize
17     \stoptext
18     


\47>:5 ->$\diamond $
                    
\symb_place_indeed ...name \??symbol #1\endcsname 
                                                  \relax \endgroup 
\symb_place_retry ...name \symb_place_indeed {:#1}
                                                  \else #1\fi 
\symb_place_normal_c ...lace_retry \currentsymbol 
                                                  \fi 
\symb_place_normal_b ...else \symb_place_normal_c 
                                                  \fi 
<to be read again> 
                   \else 
...
l.14        \item O
                   ne more column for good measure
Sorry, but I can't typeset math unless various parameters have
been set. This is normally done by loading special math fonts
into the math family slots. Your font set is lacking at least
the parameter mentioned earlier.

! Math error: parameter \Umathquad\displaystyle is not set.

system          > tex > error on line 15 in file bullet-problem.tex: Math 
error: parameter \Umathquad\displaystyle is not set ...

 5     \mainlanguage[ja]
 6     
 7     \setmathfont[modern]
 8     
 9     \definesymbol[5][$\diamond$]
10     
11     \starttext
12       \startitemize[5,columns,unpacked,two,broad]
13            \item 此の行にbulletを打つ。
14            \item One more column for good measure
15 >>         \item 此れは何処に置かれるでしょうか
16        \stopitemize
17     \stoptext
18     


\47>:5 ->$\diamond $
                    
\symb_place_indeed ...name \??symbol #1\endcsname 
                                                  \relax \endgroup 
\symb_place_retry ...name \symb_place_indeed {:#1}
                                                  \else #1\fi 
\symb_place_normal_c ...lace_retry \currentsymbol 
                                                  \fi 
\symb_place_normal_b ...else \symb_place_normal_c 
                                                  \fi 
<to be read again> 
                   \else 
...
l.15        \item 此
                     れは何処に置かれるでしょうか
Sorry, but I can't typeset math unless various parameters have
been set. This is normally done by loading special math fonts
into the math family slots. Your font set is lacking at least
the parameter mentioned earlier.

Missing character: There is no ⋄ (U+22C4) in font ipaexm!
! Math error: parameter \Umathquad\displaystyle is not set.

system          > tex > error on line 15 in file bullet-problem.tex: Math 
error: parameter \Umathquad\displaystyle is not set ...

 5     \mainlanguage[ja]
 6     
 7     \setmathfont[modern]
 8     
 9     \definesymbol[5][$\diamond$]
10     
11     \starttext
12       \startitemize[5,columns,unpacked,two,broad]
13            \item 此の行にbulletを打つ。
14            \item One more column for good measure
15 >>         \item 此れは何処に置かれるでしょうか
16        \stopitemize
17     \stoptext
18     


\47>:5 ->$\diamond $
                    
\symb_place_indeed ...name \??symbol #1\endcsname 
                                                  \relax \endgroup 
\symb_place_retry ...name \symb_place_indeed {:#1}
                                                  \else #1\fi 
\symb_place_normal_c ...lace_retry \currentsymbol 
                                                  \fi 
\symb_place_normal_b ...else \symb_place_normal_c 
                                                  \fi 
<to be read again> 
                   \else 
...
l.15        \item 此
                     れは何処に置かれるでしょうか
Sorry, but I can't typeset math unless various parameters have
been set. This is normally done by loading special math fonts
into the math family slots. Your font set is lacking at least
the parameter mentioned earlier.

columns         > balanced in 1 step(s)
fonts           > math: unset for global bodyfont simplefonts:2 at 12pt
backend         > xmp > using file 
'/usr/local/texlive/2012/texmf-dist/tex/context/base/lpdf-pdx.xml'
pages           > flushing realpage 1, userpage 1
 )</usr/local/texlive/2012/texmf-dist/fonts/truetype/public/ipaex/ipaexm.ttf>

system          > files > start used files

used files      >    1: filename=bullet-problem.tex | 
foundname=bullet-problem.tex | usedmethod=direct
used files      >    2: filename=cont-new.mkiv | filetype=tex | 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/base/cont-new.mkiv | 
usedmethod=database
used files      >    3: filename=bullet-problem.top | 
foundname=bullet-problem.top | usedmethod=direct
used files      >    4: filename=lang-us.lua | filetype=lua | 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/patterns/lang-us.lua | 
usedmethod=database
used files      >    5: filename=t-simplefonts.mkiv | filetype=tex | 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/third/simplefonts/t-simplefonts.mkiv
 | usedmethod=database
used files      >    6: filename=t-simplefonts.mkiv | filetype=tex | format=tex 
| 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/third/simplefonts/t-simplefonts.mkiv
 | usedmethod=database
used files      >    7: 
filename=usr/local/texlive/2012/texmf-dist/tex/context/third/simplefonts/t-simplefonts.mkiv
 | 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/third/simplefonts/t-simplefonts.mkiv
 | usedmethod=qualified
used files      >    8: filename=t-simplefonts.lua | filetype=tex | format=tex 
| 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/third/simplefonts/t-simplefonts.lua
 | usedmethod=database
used files      >    9: filename=ipaexm | filetype=ttf | format=ttf | 
foundname=/usr/local/texlive/2012/texmf-dist/fonts/truetype/public/ipaex/ipaexm.ttf
 | usedmethod=database
used files      >   10: filename=lm.lfg | filetype=tex | 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/fonts/lm.lfg | 
usedmethod=database
used files      >   11: filename=lm-math.lfg | filetype=tex | 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/fonts/lm-math.lfg | 
usedmethod=database
used files      >   12: filename=lm-math.map | filetype=map | format=map | 
foundname=/usr/local/texlive/2012/texmf-dist/fonts/map/dvips/lm/lm-math.map | 
usedmethod=database
used files      >   13: filename=lm-rm.map | filetype=map | format=map | 
foundname=/usr/local/texlive/2012/texmf-dist/fonts/map/dvips/lm/lm-rm.map | 
usedmethod=database
used files      >   14: filename=mkiv-base.map | filetype=map | format=map | 
foundname=/usr/local/texlive/2012/texmf-dist/fonts/map/pdftex/context/mkiv-base.map
 | usedmethod=database
used files      >   15: filename=ipaexm.ttf | filetype=ttf | format=ttf | 
foundname=/usr/local/texlive/2012/texmf-dist/fonts/truetype/public/ipaex/ipaexm.ttf
 | usedmethod=database
used files      >   16: filename=lpdf-pdx.xml | filetype=tex | 
foundname=/usr/local/texlive/2012/texmf-dist/tex/context/base/lpdf-pdx.xml | 
usedmethod=database

system          > files > stop used files


system          > structure > start used structure

used structure  > text: bullet-problem

system          > structure > stop used structure


mkiv lua stats  > used config file          - 
selfautoparent:/texmfcnf.lua;selfautoparent:/texmf/web2c/texmfcnf.lua
mkiv lua stats  > used cache path           - readable: 
'/usr/local/texlive/2012/texmf-var/luatex-cache/context/210a96fef5fbab446fd11afa77acb026'
 (order 1)
mkiv lua stats  > used cache path           - readable: 
'/home/usr/.texlive2012/texmf-var/luatex-cache/context/210a96fef5fbab446fd11afa77acb026'
 (order 2)
mkiv lua stats  > used cache path           - writable: 
'/home/usr/.texlive2012/texmf-var/luatex-cache/context/210a96fef5fbab446fd11afa77acb026'
mkiv lua stats  > resource resolver         - loadtime 0.312 seconds, 2 scans 
with scantime 0.000 seconds, 0 shared scans, 16 found files, scanned paths: 
/home/usr/.texlive2012/texmf-config /home/usr/texmf
mkiv lua stats  > stored bytecode data      - 303 modules, 65 tables, 368 chunks
mkiv lua stats  > cleaned up reserved nodes - 39 nodes, 9 lists of 446
mkiv lua stats  > node memory usage         - 3 glue, 2 penalty, 13 attribute, 
28 glue_spec, 4 attribute_list
mkiv lua stats  > node list callback tasks  - 6 unique task lists, 6 instances 
(re)created, 1208 calls
mkiv lua stats  > used backend              - pdf (backend for directly 
generating pdf output)
mkiv lua stats  > loaded tex modules        - 1 requested, 1 found 
(*-simplefonts), 0 missing
mkiv lua stats  > loaded patterns           - en::2 ja::47
mkiv lua stats  > jobdata time              - 0.000 seconds saving, 0.000 
seconds loading
mkiv lua stats  > callbacks                 - 1307 direct, 1384 indirect, 2691 
total
mkiv lua stats  > randomizer                - resumed with value 
0.98610818478563
mkiv lua stats  > lxml preparation time     - 0.000 seconds, 0 nodes, 15 lpath 
calls, 0 cached calls
mkiv lua stats  > positions                 - 3 collected, 3 regulars, 0 
regions, 0 unresolved regions
mkiv lua stats  > result saved in file      - bullet-problem.pdf
mkiv lua stats  > loaded fonts              - 1 files: ipaexm.ttf
mkiv lua stats  > fonts load time           - 0.578 seconds 
mkiv lua stats  > math processing time      - 5.422 seconds 
mkiv lua stats  > luatex banner             - this is luatex, version 
beta-0.70.2-2012062812 (tex live 2012)
mkiv lua stats  > control sequences         - 36305 of 65536 + 100000
mkiv lua stats  > current memory usage      - 41 MB (ctx: 42 MB)
mkiv lua stats  > runtime                   - 8.594 seconds, 1 processed pages, 
1 shipped pages, 0.116 pages/second

Regards
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