We purchased our new home last September and had to deal with over 1000 square feet of saltillo tiles in terrible looking condition. The dirt is mostly in the sealer, but the tile is so porous it is kind of tricky to clean. Some of our tile was black.
I used wax (sealer) strippers that I purchased from a professional janitorial supply and use a heavy commercial scrubbing machine with black scrubbing pad. As the sealer softened and came up, I added water to keep dirt in suspension and quickly mopped and neutralized the cleaned sections. It took several passes with the commercial equipment to get it. Areas under the counters have to be scrubbed with a smaller scrubber pad, and you can get something that looks like a sanding pad on a pole to minimize the bending and knee strain.
For refinishing I used a saltillo tile sealer (Royale) purchased from a saltillo tile manufacturer / supplier, and finished with a polish. You must use products specific to saltillo to get the right results. If you need the name or phone number of a supplier in CA post back. They will ship.
Lots of work, but the results are worthwhile. PS Don't use rubber backed carpet on this tile, it will leach the oil into the sealer.