I have a sort-of related question for you - when was the last time you sealed your shower floor tiles? If it's been a long time, that may be the issue. You will need something pretty strong and a lot of elbow grease to get them clean. Then, reseal them. Once they are sealed (which should be done annually), they will be easier to clean with mild products.
ETA: Oh, and alkaline cleaners to get it clean (not acidic) (as an example, bleach is alkaline, vinegear is acidic). And once it's sealed, neutral cleaners (Bona makes one, so do many other brands).
ETAA: I'm an enthusiastic user of vinegar as a cleaner, it's all I use in the kitchen. BUT - we just had our shower tiled, and the shower guy was most insistent that vinegar should never never be used on natural stone tile. Maybe it's ok on ceramic or porcelain tile, I don't know. But never on stone tile.