We know more about this topic than one might think. See this paper on Umami and the Foods of Classical Antiquity which discusses the production and economics of Roman fish sauces, particularly the wildly popular Roman condiment garum.

The typical Roman diet was in many ways similar to the so-called modern Mediterranean diet, consisting of a healthy combination of animal and vegetable proteins, vegetables, legumes, and fats (olive oil was the staple). However, it appears Romans had a greater fondness for 'umami' or earthy flavors than many Westerners today, more akin to what you would find in contemporary south Asian cuisine.

Garum was an ever-present condiment used to flavor everything from meat dishes to desserts. It was produced by harvesting the oil from a mixture of decomposing fish and salt, left in the sun for several months to ferment. Garum was high in MSG, accounting for its addictiveness, despite a flavor that Seneca describes as "salty rottenness." It was also traded widely throughout the Mediterranean as a manufactured commodity.

Although Romans did not have all of the foods that we enjoy today, including tomatoes and spinach, they did consume many of the foods identified today as containing significant amounts of natural umami substances. These include vegetables such as carrots, mushrooms, cabbage, asparagus, peas, and onions and such fruits as grapes and apples.

Pork and chicken were favorite meats, although Romans also ate beef. Seafood included sardines, mackerel, tuna, oysters, and prawns.

Our best source ... is the cookbook attributed to Apicius. It is the only cookbook surviving from the Roman world, and contains ~465 recipes.

Some recipes from the aforementioned cookbook can be found here and here.

Edited: for rule compliance.