I've used my Visa Check Card/Debit Card for each of our 4 DisneyWorld trips over the last 4 years with no problems at all. It is accepted for both normal payment and for the charge to room feature.
The only real gotcha was that once we had totalled $1000 in purchases against the room, they wouldn't let you charge any more until they charged that balance to your bank account to clear it. Then, everything started fresh again. If you find yourself in the position of being denied the charge on your card, have the CM call your resort and they will take care of all of these actions so you can continue with your purchase. You sometimes have to ask the CM to call although they should know this. Some will just hand it back and ask for alternative payment.
As advised above, I keep all my receipts, and get a printout from the front desk daily so I can balance the activity. This is definately worth the 15 minutes each night as opposed to a very confusing (and thick) bill on the last day.
The resorts in California will also take a debit card for payment, but they don't allow any charge to room option like DisneyWorld.