Tile floors are extremely common here in Florida. It is important to use only "safe" cleaning products on them - nothing harsh. The system I use is this:
I use a Swiffer mop, the kind you can put wet or dry pads on (which they sell and are VERY expensive!). Instead of their pads, I use ordinary white washcloths, which hook right onto the mop head. I dampen the washcloth and spray on a cleaning agent. I mop a smallish area, and when the washcloth is still white, it's clean! Many of my clients LOVE this system, because it shows how much I care about cleaning their floors.
Personally, I think that using a traditional type mop is a huge waste of time, because you are just swishing dirty water around on the floor.
Steam mops are great - especially if the floor is highly soiled. But with ordinary cleaning jobs, my system works great.