Two words -- steam mop.
Now, the remainder: I have ceramic tile and quartz tile...have used the sponge mop / rag mop / bucket of water with "you name it" and it was always a MESS! Got the steam cleaner ... problem solved. Be sure to use distilled water. The floor and the grout look great.
When my grandson was a little one crawling, and even now as he's 2 1/2 and quite stable on his feet, I always steam the tiles before he arrives. That way, I know he won't get any cleaning product residue on his sweet little hands. Mud the dogs dragged in, perhaps. But no cleaning product residue.
BTW, I have an Australian Aussie and a mutt, neither of which I can train to wipe their feet before they come into the house. Ha!