Unfortunately, a mop is not the best choice for cleaning tile floors. It’s biggest downfall is redepositing dirty mop water into the grout joints where it dries and leaves your grout dirtier than when you started. If you really want to use a mop, grout brush, or other method that uses a lot of water/cleaner, then the use of a wet vac immediately after scrubbing becomes your best friend by removing the dirty water from the grout surface...though a heavy, cumbersome friend, at that.
Otherwise, the Swiffer-type of damp towelettes are better for tile floors, as they clean the tile, while leaving the grout joints alone.