Dh is from Istanbul and we eat a lot of food that is familiar to him, which means we eat a lot of foods.
I don't really find it very low-fat. Salads are *swimming* in oil, for example. They just use nothing but olive oil, so it's all healthy fats. Much of what they eat that are swimming in olive oil are these cold mezes (appetizers) and I absolutely cannot *stand* the feel of lots of cold olive oil in my mouth. Dh loves it, though!
I'm not sure what the "north american" version of the Mediterranean diet is, but dh's family eats a lot of white grains (white rice and white bread is served twice a day), fried foods, cheese, lamb... but they eat a TON of vegetables (floating in fat, but still) and the meat is not usually the star of the meal, but a part of a larger whole. So for example, a meal that we make a lot here is green beans and tomatoes that are cooked with just enough ground meat to flavor the dish. Again, it has a lot of olive oil in it, though. They eat lots of beans. Beans in everything.
If a meat is the star, it's fish. They eat a lot of fish. The fish, though, is so fresh that what you buy for dinner that night was caught that morning and often is still alive and swimming in a bucket when you buy it and have it cleaned. In the US, they don't know the meaning of fresh. The "fresh" fish you get here has been on ice on the boat for weeks and then refrozen and is fresh because it's been thawed only 2 day. Blech. (Can you tell I live in the land-locked midwest?

It's probably a little better on the coasts.)
Breakfast is usually a simple affair... feta cheese, olives and maybe boğaça (a type of white bread that uses yogurt for leavening). Tea is the beverage of choice at breakfast. They like cured meats and will eat them for breakfast on weekends (like sausage, called sucuk). They also like offal (nothing is wasted). They drink turkish coffee mid-morning and after supper. None of this sweet, calorie-laden Starbucks. It's just coffee.
Every single meal ends with fruit and sometimes nuts (which may be served with after dinner drinks - scotch for the men and coffee for the women). Quite literally, dh eats 4+ servings of fruit after lunch and dinner both. They don't snack. They'll have a midafternoon tea (around when westerners eat "dinner"... 5ish) that might include a bite to eat, but they keep it really small. They eat supper late - like 8ish.
So, that's my experience with it. I'm sure Greek, Italian, Maltese, Cypriot, etc. diets are different. We eat "mediterranean" because it's what dh grew up with and I enjoy cooking it, but we don't follow it as a "diet", but as a familiar cuisine that is part of our overall diet. I almost never see an overweight Turkish person. They're doing it right, that's for sure. But it's far from low-fat. I'm not sure what they really eat is the same as what is marketed to be "mediterranean" in the west. Rather than following what Americans say about it, perhaps the best plan is to get authentic cookbooks from the region and try some of the recipes. Just a thought. Wanted to add... the library where I work has some really great cookbooks of the region and most libraries try to have cookery books of all ethnic cuisines, so that would be a good place to start for the cookbooks.