Cleaning Marble Gravestones
One of the most common questions we receive is about cleaning marble gravestones. While people generally think of gravestones as tough and hard, that is not the case with marble gravestones. Marble can be carved with wood-carving knives. Marble is beautiful because it can be carved. It is also porous. So any cleaner you use will seep into the stone and will likely leave residue.
Unfortunately too many "experts" recommend bleach. DON'T USE BLEACH! Household bleach is sodium hypochloride, and it is the sodium that causes the problem. Marble gravestones are calcium carbonate, and the sodium in the bleach reacts chemically with the calcium in the stone to form a salt -- eating away calcium from the gravestone. Unfortunately, the damage is not immediately apparent; so household bleach is frequently recommended by well-meaning folks as a quick and effective cleaning method. Afterwards, during every rainfall or lawn watering, particles of the gravestone are simply washed away and the stone gradually yellows. 
Don't use detergents, soaps, vinegar, bleach, or any other cleaning solutions on the stone, no matter how mild! Wire brushes also will permanently damage the stone -- use a nylon-type brush with the consistency of a toothbrush. Do not use pressure washers on marble stones -- you will wash years of the stone's life away and damage the smooth surface of the stone. Once the surface is damaged, dirt will more quickly accumulate.
Using a nylon brush and lots of clean water will remove most of the dirt from a marble gravestone with minimal damage. Only when lichens or other harmful dirt is on the stone, should it be cleaned. Chicora Foundation is among the top in the field on gravestone preservation. Here is their web page on cleaning gravestones: http://www.chicora.org/cleaning.html
Here is a summary of the study that was done by the National Park Service on good cleaners for marble gravestones that have biological growth: http://www.cem.va.gov/hist/hspjtdesc.asp but basically it says the 3 that are the best, as known at the current time, are D2 (D/2) Architectural Antimicrobial, Daybreak, and World Environmental Group’s Marble Cleaner. These do the least damage. The final and full report is dated July 2011: Final NCPTT report on cleaning gravestones (PDF file).
Frankly, we use water and a brush. When we feel we must clean a stone that has biological growth, we use D/2 available from LimeWorks.us at http://www.limeworks.us/blog/index.php/d2-biological-solution/ or the Cemetery Preservation Supply, LLC at http://www.gravestonecleaner.com/. It can also be used on granite stones.