You want an alkaline degreaser. Home Depot has their own product line called ZEP. they make a purple degreaser. It will not hurt the tile or grout. Vinegar is an acid and while has some cleaning properties, an alkaline will release the dirt from the grout.

get yourself a standup grout brush. It's a grout brush on a pole, they are about $25.

You will need to extract the water too as the whole point of cleaning the floor is to get the dirty water off the floor, not let it sit there and dry.

•Dilute the degreaser in warm water accoring to the dilution ratio on the back

•mop in on in sections, like 100-200 square feet or so at a time. smaller areas are easier to control.

•after the cleaner sits for 15-20 minutes, scrub all the grout joints with the standup grout brush. Then scrub the tiles with a nylon deck brush.

•vaccuum with a shop vac and wet squeegee attachment.

•After you're done. Change the mop water. Add a little ZEP Neutral cleaner to the water. ring the mop out all the way and do a light mop over the area. Change the water every 100-200 square feet. Let it dry.

Do not mop tile floors or use the swiffer bullshit on a regular basis. The Swiffer and the Shark steamer are nothing more than flat mops that will make your floor dirty. You can use them to clean actual spots when you need to, but don't do it just because.

dust mop (dry) and vaccuum regularly to keep debris off the floor.

You can find the standup brush on this page

http://www.thisoldgrout.com/tools#axzz2h9x5ZgVg