You can take a backpack on RnRC just fine. You just put it at your feet (though there's more room for it in the front row of each car than the back row).
Off the top of my head, there are no attractions that prohibit you from bringing a bag on the ride. They have lockers outside Kali River Rapids, but that's more of a convenience to keep your stuff dry.
That being said, there *are* a few attractions where you'd have trouble with a larger bag:
-Space Mountain has very limited room at your legs. Rows 1 and 4 have a little bit more room for a bag.
-Mission Space has a compartment in the capsule for your bag, but a larger backpack won't fit. I don't know what they have you do if it doesn't fit, but you'd probably just put it at your feet and make sure to keep a foot on either side.
-Soarin, if the bag won't fit in the pouch under your seat, they'll have you put it off to the side where they park the wheelchairs and scooters.
-Tower of Terror, just loop the strap around your leg and you'll be fine. Same with Dinosaur and Expedition Everest. (I think there's even a metal hook above the storage pouch that you can hook the bag's strap onto to help)
-Astro Orbiter, there's not much room for people inside the rocket, let alone a backpack. You can probably shove it up toward the front of the rocket, or just leave it on the ride platform off to the side and walk around and grab it when your ride ends. (I've heard that after the upcoming refurb rockets will always stop right where they started so that they can have a wheelchair transfer rocket)
-Rob
