I've taken my SLR cameras on every ride I've ever done at WDW.
Keep in mind that the vast majority of rides at WDW are not thril rides. Stuff like Pirates, Haunted Mansion, the Great Movie Ride, and Spaceship Earth are quite tame and perfectly safe for any kind of camera. Only a few rides require special attention to keep your camera safe and/or dry.
For coasters, like Rock and Roller Coaster or Big Thunder Mountain, I typically hang the camera by the strap between my feet.
Some thrill rides, like Star Tours, Dinosaur, and Mission: Space, have storage areas that will hold an SLR while you ride.
For water rides like Kali River Rapids (or periods of heavy rain), I use a
home-made camera poncho. It works perfectly - I'd have to completely immerse the camera for several minutes to get any serious moisture in there. And I normally don't bother with the ponch on Splash Mountain, but I do shield the camera with my body on the last drop, just in case. For some reason, I just don't seem to get all that wet on Splash. Usually.