Interesting. I don't call those "Mediterranean," for me they're "Middle Eastern."
Dh is from Turkey and he grew up right on the Mediterranean Sea... Turkish is the quintessential "Mediterranean" diet. I know of nothing more "Mediterranean" than Greek and Turkish food, actually. Like any food, the culture of what people eat doesn't start or stop at a border... it's a continuum. I'm sure the food is different in the north and east of Turkey closer to Iran, Georgia and Armenia than it is in Istanbul. Still...
In Turkish appetizers are called "Mezes". They usually put out cold appetizers that will sit on the table for the entire 3 or 4 hour(!) meal. The hot appetizers are brought out one by one.
Börek (hot app)
Saute together 1/2 lb. ground beef, 1 diced onion, 1 tomato in some butter. Cool then add 1 egg, a cup of chopped parsley, some dill and salt and pepper. Mix together. Using two phyllo sheets, brushed with butter, cut into long strips about 2" wide. Add a small bit of beef mixture at the top of the phyllo strip, then wrap with the "folding a flag" method (like spanikopita triangles), brush outside with butter and bake about 25 minutes in a 350/180C oven. (I can get more detailed if you like this idea, but there are tons of recipes on the net that even have pictures.)
Mücver (hot app)
3 shredded zucchini
salt
3 eggs
5 scallions (green and whites) chopped
1 c. chopped dill
1 c. good french (sheep's milk) feta
3/4 c. flour
2 t. baking powder
oil for frying
Shred zucchini. Salt and let sit in a colander to drain for a couple of hours. Squeeze out excess liquid. Combine remaining ingredients except oil. Form into patties (like crab cakes). Fry in oil. Salt as soon as they come out of the oil. These work REALLY well at room temperature. They are a favorite here.
Typical Bean Salad (cold app)
2 cans Cannellini Beans, drained (or dried, cooked)
1/2 red onion sliced very thinly
2 roma tomatoes, seeded, chopped
1 c. chopped cilantro (or parsley if cilantro is not liked)
good extra virgin olive oil
lemon juice
Salt and Pepper
Another cold app that they like to do is a variety of fresh vegetables dressed simply with REALLY GOOD extra virgin olive oil and freshly-squeezed lemon juice.
If you google for turkish mezes you will get a boatload of very authentic recipes. In Greek I think they're also called Meze or Mezedaki or something similar. I'd also look up Spanish tapas.
Good luck!