Mopping the traditional way generally ends up just spreading around dirty mop water. Using the system we recommend will be much more effective. Grout lines will stay clean longer also.
First, you'll need the proper mop. Lowes sells a Rubbermaid Microfiber Mop that will work well for you. Buy one of these and a couple extra mop heads.
Second, you'll need a neutral ph, no-rinse cleaner. Both Lowes and Home Depot have these available.
Mix the cleaner according to directions(more product is NOT better). I like to mix it up in a spray bottle. I spray about 100 square feet at a time and then mop it up. A light mist should work just fine if you clean fairly often. After spraying, just wipe it up with the microfiber pad. Change the pads once they start to show significant soiling.
This is the method that has worked best for me. It prevents slopping around dirty water and doesn't leave a film on the floor. Over-wetting and not changing out mop water often enough is the most common reason for dirty grout.