While there were many similarities between modern diets and the ancient Greeks and Romans, there were also many differences.
Surprisingly, the Romans rarely rotated butchered meat into their diets (cows were very rare, pork was not common). They would instead eat things like game, seafood, and poultry. They hunted for much of their meat, they would use oyster farming and utilizing the fact they were on a peninsula, while poultry included duck and geese. The Romans also actually farmed for snails, a practice that seems foreign to us. The Greeks were far more likely to eat meat, and commonly ate beef.
The fruit and vegetables consumed had a lot of crossover between what is eaten now. But just because two things have the same name, does not mean in any way that they are the same plant 2000 years later. Some of the staples of Italian diet’s today were not introduced until later trade with Asia, the Middle East, and the New World. Some of these were eaten then, and are technically eaten now, but the nutrition it provides, the color, the taste, and the consistency are all different. Pliny the Elder had 40 different types of olives to access alone. But some of these plants were driven to extinction, or fell out of favor.
Some things were actually eaten both then and now, but are used in completely different ways and held in completely different levels of esteem. Eggs and cheese were considered food for the upper class. Their bread was commonly consumed by dipping it in wine. And oranges and lemons were not considered food for cooking, but food that could be used for medicine, especially fighting scurvy and providing vitamin C and anti-oxidants. This idea was actually common, and has some merit to it. They saw diet as the building block and the best medicine. While sometimes medicine can help, it is true that your diet can have profound positive and negative impacts. They may not have had a full grasp on medicine, but they did understand cause and effect. And when you do not change your food from its natural state too much to begin with, you tend to have less health problems related to processed food. High cholesterol, obesity, diabetes, the list goes on. When you prevent the problems from happening, they do not surface.
The Romans traditionally had a very small breakfast, a light lunch, and one large meal of the day around dinner, and a possible snack afterwards. This type of consumption is much more in line with what some anthropologists believe early man ate like. Eating small meals of nuts, plants and roots the women collected during the day, and big meals when the men returned from the hunt at the end of the day with game. Obviously Rome did not have this kind of scarcity. But it is possible they got their traditions there. The set schedule of meals would not be the same or as consistent if you were not upper class, however.