you don't have to be wearing sneakers - we do the flip flop thing too. (actually we each take at least 2 different pairs/styles of shoes and change every day or two so we don't get foot rub in the same spots and blister.)
most rides your type of footwear won't matter much at all. on rides like soarin for example, if you have any type of shoe that is open in the back (does not have a back or an heel strap) they will ask you to take them off as you are sitting in the seat, before the ride takes off. they are sitting there waiting for you when you land and you can slip them back on again.
not sure about other (non WDW) parks that might have the inverted harness type coaters where your feet would dangle. i would have to assume that they have those cubbie hole type shelves where you can slip off your flip flops, tuck them in a cubbie and then recover them when the ride comes back.