We have marble border stripes on the tub surround in our MB, and split face travertine border tile on the tub surround in our hall bathroom. Been using 7th Generation Disinfecting Bathroom Cleaner (spray). I spray it on, let sit a few min., and then wipe dry with a microfiber cloth (no rinsing necessary). I usually spray some on the microfiber cloth before wiping in order to help spread it around on the tile and grout lines.

We also spray Mrs. Meyer's Shower Cleaner (the daily spray) after showering.

I called both companies to make sure those products were natural stone-safe. Some others from both lines are not.

I use Marbalex (from SCI) mixed with water for mopping all of our floor tile--travertine and porcelain. It's a no-rinse stone-safe floor cleaner. I don't know if it disinfects, but we are fine with just using soapy water to clean our floors.