Are you around most of the time, so you can react quickly to "accidents"before they can seep in? Do the furkids have the run of the whole house all the time? Maybe you could restrict their access to certain rooms when no-one is home. For example I don't let Ally and Connor into the bedrooms or lounge (which are carpeted) unless I am with them, as the rest of the house is tiled.

I have to agree with Pops - I LOVE tiles. They are hard-wearing and really easy to keep clean. Also if you are worried about the rooms being cold in winter you can always get under-floor heating.