Greek medicine, which had significant impact in Rome as well, was very heavily focused on diet. Aristotle, both a natural philosopher and an ethicist, recommended moderation (not abstinence or extremism) in diet and in other pleasures as a critical part of the state of well-being. Hippocrates advocated diet as well as exercise, bathing, and other preventative measures that would give the body strength to survive and resist disease. Galen, a Greek physician in the Roman Empire, codified the system of four bodily "humours" with which you might be familiar, which continued to have tremendous impact on European medicine for at least a thousand years after his death. In this system, proper health requires a balance of heat and cold as well as wetness and dryness that is dependent on environmental factors, especially food. Thus, a wet cough might be seen as indicative of cold and moisture which required the consumption of something hot and dry to improve.

I'd say, then, that there wasn't overall so much a sense of "counting" which has become so popular in modern diets as in balance or harmony -- a proper mean between indulgence and temperance, between different complementary sorts of food, etc.

I'd recommend sitting down with the original authors if you can bear it -- Galen in particular is dominant in Roman medicine, and his text "On Food and Diet" will certainly have what you are looking for. Helen King wrote an overview which I was believe titled "Greek and Roman Medicine" which was the shortish text I recall using in grad school.