If you are talking about travertine floor tiles, the black marks are dirt stains that have set into the stone. It can be particularly challenging to remove. I would begin by using very hot water and a nylon scrub brush. If that doesn't work then I would use a product called Stonespecific Stain Remover by Custom Building Products which you can pick up at Home Depot. What I like about this product is that you don't have to do any scrubbing, just let the product do the work. Be sure to follow all safety and application instructions and do a test spot in a hidden area to make sure it does not etch your floor.  After you get the floor cleaned be sure to seal the floor with a penetrating sealer appropriate for travertine.

Here is a link to the cleaning product:

www.custombuildingproducts.com/Pro...

Also, be sure to check out our stone floor care guide for more information:

www.findanyfloor.com/stone/StoneFl...