It sounds like you have done everything you could. The next step is to get more aggressive with your tile. You may want to hire a pro to do the job but if you insist on doing it yourself I would suggest using a solution of sufamic acid crystals and water. Follow the mixing instructions on the container. Make sure you use safety goggles, rubber gloves, and adequate ventilation. Always do a test spot in a hidden area first to make sure it is not bleaching out your grout joints nor etching your tile surface. Dampen your floor with plain water, then apply the acidic solution to your floor, allowing it to soak a few minutes, then follow up with an acid resistant scrub brush. Mop up the acidic water then rinse with clean water a couple of times. This should do the trick.

 

Another product you can try is called Bar Keepers Friend which is like Comet on steroids. It is slightly abrasive, so again, do a test spot. I would dampen the floor, then sprinkle on the powder, then use a nylon scrub brush to work it into the surface.

 

Also, be sure to check out our tile care guide:
www.findanyfloor.com/tile/TileFloo...

 

Thank you Michael.  You have been most helpful.  I tried Bar Keepers Friend as it sounded like the easiest thing to do first.  Your description of comments on steroids was absolutely right!  It did the trick, for the most part.  There are a few of the deeper lines that are still holding dirt, but my floor looks 99% better.  

 

Clean your marble stone surface with a soft cloth or mop using pH neutral cleaning solution. Always wipe off any spills immediately as they happen. Never, ever clean this floor with vinegar, as you will destroy the finish. The easiest and BEST way to clean granite is to buy a daily STONE cleaner. I reccomend you to use Lustro Italiano Cleaner and Polish specifically designed and PH balanced for Marble, Granite, and Dimensional Stone Surfaces. you cna find it here www.ciciliotandson.com/products.php