Vinegar can cut through a lot of nastiness on a tile floor. Also, if it's REALLY bad, get a deck brush (it looks like a push broom, but the bristles are stiffer). That is what restaurants use to deep-clean their kitchen floors. At a certain point, mopping won't remove any deep-set dirt, but a deck brush will loosen it up, and then you can mop it away. If (god forbid) you have light-colored grout, try the Clorox Bleach Pens. At our last house, I realized that the grout in my entry way wasn't supposed to be charcoal gray... used the Clorox Pens and a deck brush, and voila! My grout was white again! (Who in the world uses white grout in an entry way?! I was so happy to move out of that house...)