In reply to matt davies:
Porth Ysgo is great. It's a bit of a drive, but its location means it is often dry when the mountains are wet.
Parisellas cave is often dry when it is raining. It's very steep, very polished, surrounded by tourists, and the cave is full of sheep/goat poo. People tend to love it or hate it.
Other potentially dry limestone options are The Gop, Pnatymwyn and tremerchion.