I think there was some discussion awhile back about using other touch screens (i.e. iPad touchscreens). I am not sure what this would involve, but you would have to take a look at the specification sheets to make sure that the right voltage/wiring is done to the Arduino (and whether it can handle the increase in inputs that would be required for two-touch, etc). Don't forget that the existing libraries would be unusable, and you would have to write/find them.
Using a non-standard LCD would be more difficult than using the LCD and LCD shield that have already been optimized for the Arduino.