Donating your body is a much bigger process than organ donation -- Cadavers used for medical research or education go through almost a year of various chemical processes before arriving in a cadaver lab. Also, you can be a registered organ donor and not be able to donate your body -- we ran into that with my mom. She'd wanted her body to be donated to MS or Behcet's research, and I bet they'd have learned a TON from her, but they couldn't take her because she wasn't specifically registered as a body donor through them. It's asinine, but there it is.

Also, AFAIK, once you've donated organs, unless you have a disease of some sort that would make your body useful to specific research, med school labs can't take you because they need each cadaver to have teaching points for all internal structures, wherever possible. (although they've got women who had mastectomies/hysterectomies, and men who've had bits removed, etc -- it serves as a teaching tool for whatever disease process caused it, and as a diagnostic -- what's missing in this picture, that sort of thing.)

Check with the institution that interests you most, and ask them their guidelines. Cadavers are seriously expensive business, so if you offer to send along a grant with your deceased person that would help cover the cost of the process, they'd probably kiss your feet. (or...so to speak. LOL)