I have a similar problem, stone floor in my shower and I'm not sure if it's marble. I've found this on line, but haven't used it.

Here is a link that might be useful: Marblelife

Since marble develop stain easily, so clean the marble and shower surrounds when leaving wet glasses or shampoo bottles sitting too long. Do wash the tiles once a week using a neutral pH cleaner as they are really effective in cleaning process. However, if your marble tile is extensively stained or etched, you may have to call in a company that specializes in marble restoration.

The best way to clean marble is to wipe it with a damp cloth using a mild detergent and then wipe it dry. Also for tough stains, try a Salt and a lemon half. The salt will act as an abrassive and the slight acidity from the lemon will bring deposites out from the marble.

Is this "real marble" or "cultured marble".

Will make a difference in how you clean it.

Cleaning of marble requires extra potential and maintenance, care. You can use most effective, cheap and simple DIY solutions to clean all types of marble, you can use cleaning agent, soft cloth, hot water and specially formulated stone cleaner to clean the marble. You have to sweep and vacuum the marble regularly to let it clean and remove dirt or small rocks.

I use a steamer on my basket weave marble. I bought a canister type so I get a significant amount of steam lasting up to 30 minutes,