Determine Source of Stain to Find Best Cleaning Solution
As a 35 stone and tile industry veteran, it is best to first determine the source of the stain to find the best cleaning solution. For instance, the above mentioned use of Sulfamic Acid is best used for cleaning relatively new grout residue, but NOT for greasy stains. The mild acids are useful from melting away cement and grout residue.
For grease and other organic stains, mild alkali cleaners are best. And I DO suggest using professional grade cleaning products specifically made for stone and tile like HMK R55 Intensive Cleaner.
These products can save you time and can also do a better job than home remedies. Some of the options mentioned can actually damage tile (particularly marble tile) like vinegar and baking soda.
Baking Soda is pH 9.0 and can harm polished marble. Vinegar (depending on dilution) can be as acidic as 3 on the pH scale and can etch marble, travertine, limestone and can also eat away at grout, particularly unsanded grouts that are cement based.