I was completely unsuccessful. My professor was stubborn, and wouldn't let me
do my homework on paper because it would be "hard" for him to allow me to do
so, and because he knew that I had the ability to run Flash and do the
homework online. In the end, I dropped out of university after that semester,
though this decision was mainly because of the huge cost of taking classes
and the nature of student loans; I don't think I will benefit enough from a
degree to make the massive inescapable debt worthwhile, and there's nothing
in particular I want to learn at a university. (Ironically, it was one of my
classes at the university that made me think about the inescapable debt
problem, and that was incidentally the only class I liked.)
My primary recommendation based on this experience would be to show your
professor that you're committed to your refusal to use proprietary software
by either not doing your homework (if you can't even see the problems; that
might be the case, it was for me unless I ran proprietary JavaScript and used
Gnash to trick the website into thinking I had Flash); or, if you can, doing
the homework on paper and showing it to him. If he is as stubborn as mine was
and just won't let you do the homework on paper, plan to re-take the course
next semester, and start by talking to other professors who teach the course
right now and finding one who would be willing to let you do the homework
without running proprietary software. Negotiating before enrolling in the
class will probably give you better success than negotiating in the middle of
the course.
Another minor suggestion: say "libre" rather than "free". I find that people
understand "libre" better and with far less explanation, because they don't
wrongly think they know what I'm talking about in advance.