There may have been changes since I've been there, but as of July, Disneyland let me record any ride I wanted to with my GoPro using the wrist strap.
Once you figure out how to aim it at yourself and your family you can get some really incredible footage, though what I typically use it for is to film big drops or moments on outdoor rides, then use my computer to take screenshots of the best parts of the videos and print the resulting photos. Some of the frame grabs I've gotten from the big drop on Splash Mountain or the top of the loop on California Screamin' really turned out to be really great, and much better than any on-ride photo Disney has ever tried to sell me.
Disney World was even more permissive with GoPro use, at least 2 years ago or so when I was really active with it.
On one vacation I again used the wrist strap and strapped the camera to the hand bars in front of us on Kali, Everest, and Thunder Mountain. With these fixed, stable mountings, the footage was even better, and a whole lot more watchable than my wrist footage.
Since then, however, the theme park industry has really cracked down on GoPro usage, and I really doubt Disney would allow such shenanigans these days.
For the record, both Knott's Berry Farm and Universal Studios has made me put away my GoPro before using it on rides.
No Disney park ever has.