Need advice on good products for cleaning VERY dusty tile floors!?

I just moved to the dustiest place in the whole world: New Mexico. I'm usually not very finicky when it comes to 'housekeeping,' but I can't get any sort of handle on all the dust in my home and it's out of control. When my toddler splashes water on the tile floor, the dust then turns to a muddy soup. Apparently, ordinary mops don't do the job! They just drag the dirt around. Any suggestions about really good mops/floor cleaning products on the market? Mr. Clean's mop didn't do the trick, and neither did the oh-so-overpriced "Microfiber O-Mop." Thanks much!