The thing is the fact that travertine is a rock that is very porous. It soaks up blots just like the blots along with a sponge can go relatively deep down to the stone--depending on whether the natural marble stone has been sealed or not. What this means is that for some spots the lone way to get them out is to "sand" down the stone until you have removed the stained portions of stone. This is usually done using steel wool or an abrasive polishing compound. For many stains, however, there is an option that is more straightforward.
Clean the stained area with a mild dish soap and also hot water. Sponge the soap and warm water on the spot and let it soak into the spot for two minutes. Wipe using a sponge dipped in water that was plain. Blot the area with a clean cotton cloth, putting plenty of pressure on the spot.
A commercially-available cleanser particularly made for travertine. Such cleaners are available at most home improvement stores and at any store which sells stone products. Follow the Maker's directions exactly.
Pour on the spot and cover it using a part of plastic wrap. Permit the hydrogen peroxide to work with the blot for half an hour. Eliminate and sponge off the hydrogen peroxide with plain water. Blot the stain with a clean dry cotton cloth, putting pressure on the stain so that it'll soak to the fabric.
Use 00-grit steel wool in the event the spot still remains. This is going to take time. Be patient and just keep rubbing the spot with all the steel wool. The notion is to take miniature layers of rock off until you have remove the last stained layer.
Pour a half dollar-sized dollop of rock polish onto the stained region and utilize an automobile buffer running at low rate to buff the place. Again, until the last stained layer is removed, the idea would be to remove layers of stone. Use only moderate pressure on the buffer and be patient. Depending on how deep the stain is, this could take the time.
Hints & Warnings
If using rock polish and a buffer be certain to sometimes feather the region out --in other words, so that you don't create a score in your rock which will be noticeable buff a broader area than only the blot.
Never use cleaning agents which include acids, like vinegar, window cleaners or lemon scented furniture cleaners. Use cleaning agents designed for travertine.