I would recommend charging to the room and then paying with gift cards if you go to table service restaurants.
I've shared this story before, so I apologize if you've seen it already. But my DH and I ate at 'Ohana in January 2014 and I paid with a gift card. I was using the Disney gift card app to track all of my gift cards. It was a birthday trip, and several people had given me Disney gift cards for my birthday to use on the trip, so I had a lot to keep track of. Anyway, when I was verifying everything that night, something wasn't matching and it took me a long time to figure it out. Eventually, I realized that I had not been given back the same gift card at 'Ohana that I'd used to pay. I'd input all of my gift cards before the trip - the one I'd paid with at 'Ohana was no longer in my possession and the one I did have was not in my gift card tracker, so I knew it was the wrong one. It had the same design on the front as the one I'd used to pay, so I didn't think anything of it. There was less than $5 still on the card, so I didn't pursue any action over it. I felt like I'd been potentially robbed (the card I had been given back, of course, had a $0 balance on it), but again, I wasn't going to spend vacation time chasing down $4 and some change. Obviously if it had been my personal credit card, I would have known right away if it was the wrong one because the name would have been incorrect.
An alternative would be to take a photo of the back of your card or write down the number somewhere and compare when the check is brought back to you.
But to answer the OP's initial post...unless it's a very recent change, all WDW owned and operated restaurants accept Disney gift cards. There may be some kiosks that don't, but I successfully used gift cards at several kiosks on my trip.