Just came back last week and I want to share this not-so-secret place. The Esquillino market is a ten minute walk from Termini and was a great place for fresh food, frequented mainly by the local ethnic community.
If you have access to a kitchen during your stay, you must buy some of the fresh fish (some still flapping!) to cook. The prices are great too!
There is also a great variety of fruits, cooked meats, breads and exotic spices. Not sure how much of these are allowed by your countrys customs but it was a great way to spend an hour or so if you stayed nearby.