They tended to be Greek as the eastern empire was very much part of the Greek world even during the height of the Roman empire, but obviously with the influence of Latin food.

There seem to be only two books on the subject: volume five of Phaidon Kukules Life and Civilization of the Byzantines and Andrew Dalby's Flavours of Byzantium (reviewed here and here).

Supposedly the Byzantines' intellectual concept of food was based on Galen's theories i.e. food influences the various humours. As for the foods themselves, the Byzantines lacked an Apicius, so there are references to foods but no comprehensive list of recipes. The food was much like Greek food but as time passed with more and more access to different spices and ingredients. For ancient greek food there are many more sources such as Athenaeus etc...

We actually know a fair amount about the types of foods that Byzantine ate throughout the centuries. In the early centuries, foods remained similar to what they had been under the old Roman Empire, but as time went on, the Empire gradually developed its own distinctive culinary diet.

Having read several interesting modern studies on Byzantine cuisine, including: Dalby's Tastes of Byzantium, mentioned by /u/unique_pseudonym, and The 37th Byzantine Symposium's Eat, Drink, and be Merry: Food and Wine in Byzantium, I will briefly comment on the highlights of the Byzantine diet during the height of the Empire.

By the early Middle Ages, the city of Constantinople had established itself as the largest trade hub on the European continent. It received and traded exotic and mundane goods from as far away as China, but also acted as a hub for locations closer to home: the lands of the Rus, the Caliphate, and the West, to name a few. Within the Queen of Cities developed a very diverse cuisine, based on indigenous Roman and Greek dishes, but then given a twist of exotic spices and condiments from half a world away.

However, this is not to say that the entirety of the Empire ate this way. In the far-flung reaches of the Empire, the specialty ingredients found in Constantinople were generally hard to come by, but during the high points of the Empire, some of the larger cities outside of the Imperial heartland of Thrace and Asia Minor (most certainly Thessalonika, and probably Athens) most likely had vendors that sold them. For the Byzantine farmer, even cheap spices probably remained mostly out of reach, but for townsfolk and city-dwellers, they were generally affordable.

The types of foods and spices available to regular citizens can be gleaned from several sources, one of which is a law passed by Emperor Leo VI in his famous Basilika Code of Laws that recites a list of ingredients sold at grocery shops in the city of Constantinople.

"Grocers may keep their shops anywhere in the city, in the broad streets and on the city blocks, so that the necessities of life may be easily procurable. They shall sell meats, salted fish, guts, cheeses, honey, oils, fruits and vegetables of all kinds, butter, solid and liquid pitch, cedar oil, hemp, linseed, gypsum, crockery, storage jars, nails, and in short, everything that can be sold by steelyard and not by scales."

These types of markets were presumably present in every Byzantine village, town, and city, although, the many markets of Constantinople would have easily been the most diverse. Using the wide varieties of ingredients available, everyday Byzantines could go about cooking their own foods at home. However, in the larger cities, to those who could afford it, there were tavernai where one could dine out as well.

In many sources, scattered bits of information regarding the actual dishes that were served, paint a decent picture of some of the types of foods eaten by native Byzantines. I will list a few here:

Apotki

Mentioned in Constantine VII's De Ceremoniis as a food for a feast, Apotki is a type of salted pork loin, submerged in vinegar, and then rubbed with cinnamon, pepper, and the savory herb. It was allowed to cure over the course of several weeks, to lock in the flavor, and was probably served on a tray.

Monokythron

Presumably a proper home-cooked meal mentioned in the Prodromic Poems, this "one-pot" meal consists of cabbage hearts, eggs, cuts of cooked fish, a small amount of a variety of cheeses, garlic, pepper, olive-oil, and before it is served, "a splash of sweet wine" on top.

Aphraton:

Another presumably home-cooked meal, found in Anthimos' Letter on Diet, aphraton is simple dish made from chicken and egg whites, cooked as a casserole and served with the famous Romano-Byzantine garum sauce. To impress your guests, it is recommended to add some fish or shellfish on top. It is suggested to be eaten with a spoon.