For the first time use Krud Kutter. I use a swifter mop with white terry cloths rags which can be thrown in the washer and reused. Personally, I hate having to clean a cleaning machine after I've cleaned the tile. Put the Krud Kutter in a cheap, plastic spray bottle, spray a section of the floor and wipe it up with the mop/rags. Repeat, if necessary, until the rag comes up clean. After the floor is all mopped, spray the floor with vinegar and go over it with the mop for a shine.
After the first time, you can use a solution of 1/2 Krud Kutter and 1/2 vinegar for regular mopping.