Use a ph neutral cleaner. I'll guarantee you that same tile store will be able to get you one from one of the sealer companies.

I have marble tiles in my shower and there isn't a vent or fan. However there is a window and after a shower I quickly squeegee the walls and floor to remove excess water and then leave the door open. It only takes a minute to do because its a small shower. My tiles are light gray, matte finish with a lot of veining so the dirt doesn't show right away, but I try to look for it and scrub it with a brush as soon as I see dirt. Then I am careful to use only baking soda/hydrogen peroxide. If the grout needs a deeper cleaning, I steam it with a Bissel home steamer maybe 2x per year. So far, the tiles/grout look great and its been 2 years. I think the trick is to catch it early and don't let water sit. So its a little extra effort when getting out of the shower but it eliminates having to scrub a lot longer once a week and it always looks clean. And I will reseal every year. That makes it a lot easier to clean the grout too.

Some of the most gorgeous kitchen backsplashes I've seen are marbled ones and, unfortunately, it's true that this beauty comes with a price as marble is easily stained, dulled, and etched. However, you can prevent this if you learn how to properly take care of your marble backsplash. To protect it from scratches, don't use any sharp-edged objects against it. You said you're cooking Indian food. Curry powder and cumin, for example, often used in Indian cuisine, can leave stains on marble. And other strong pigments like coffee grounds and leafy greens can leave traces, so make sure to sweep them right away. Also, coffee, orange juice, and wine can easily stain marble. To prevent staining, wipe those spills as soon as possible because marble will definitely stain if those spills sit for a while untouched. The right way to clean your marble is to use some warm water, a gentle dish soap and a soft cotton cloth. Don't ever scrub marble backsplash - this would definitely scratch it. Make sure to wipe the marble backsplash dry. Otherwise, it may stain. To make it shine, use chamois cloth. If there is a stain already on your marble, make a paste of baking soda and water to a consistency of a paste. Leave it up to 24 hrs under a plastic wrap and remove it afterward. Just do some light cleaning frequently and your marble backsplash will beautiful for many, many years!

@Dragonfly - Thanks. My counters are only cleaned with product once a week but if I actually spill something on the counter, I wipe off with a damp washcloth. I do clean up any drips around the sink which occur when I wash my face but actually I do that more for the faucets than the sink as I have hard water and didn't want residue accumulating on the faucets over time. I do use a steam cleaner on the bathroom floor periodically. I have read mixed advice about this but so far it seems to be fine. I don't do anything with the shower floor in terms of cleaning but I hose it down for any suds before exiting. On the advice of people on this forum, I do use liquid shower gel which is what I would be using anyway as I used it when I had the hideous fiberglass shower/tub insert :-). Marble and natural stone can be restored by a professional. I live in a condo with marble floors in the common area. Periodically, my HOA hires a professional stone restoration company which cleans and polishes the floors and also cleans the grout. The floors then look as good as when they were installed twenty years ago.