At WDW, they let you take bags and backpacks on rides. Some rides have a specific storage area (like Soarin, which has a storage area under each seat). For other rides, you can simply put the bag on the floor by your feet, loop your leg through a strap to keep it in place, and go! (This even works on "upside down" rides like Rock n' Roller Coaster, which has inversions. Just set the bag on the floor, loop your leg through a strap, and go. Just make sure you zip up all the compartments!)
At non-WDW parks, like Universal, I think they don't let you take bags on rides but they do at WDW! I've never been on a ride where it was a problem.