The Greeks do lamb much more than they do beef. If you are willing to try lamb, you will find tons of great recipes, using ground lamb, like moussaka or meatballs, to roast leg of lamb, a delicious feast that can be served hot or cold. Most marinades for the leg include tons of garlic, oregano or rosemary and parsley, olive oil and sometimes dijon.
Also, Greeks cook a lot of fish and shellfish, dressing them simply to let the freshness come through. I really like the shrimp & feta dish with ziti and tomatoes.
Chicken cut into serving pieces, 3 large onions sliced, 3 garlic cloves minced, 4 large green peppers cut into 1/2" wide rings, 3 hot dried chilies, 1/2 cup canned diced tomatoes, ground allspice, ground cloves, ground cinnamon; plus course salt and freshly milled Tellecherry pepper. Sear chicken in butter in a Dutch oven, or similar. Remove to platter, add the onions, peppers and garlic to pan and cook till wilted. Chicken goes back in along with all the spices, the tomatoes and S & P. Water just to cover the chicken is poured in, pot is covered, heat is reduced and everything simmers for about 1 1/2 hours till the chicken falls off the bone.
For chicken, I like to marinade thighs in lemon juice, garlic, olive oil, salt, pepper, crushed red pepper, oregano, and sometimes thyme. Grill low and slow so they render a lot. While they're grilling I boil the marinade to kill all the bacteria. When I pull the thighs off, I toss them back in the marinade. Kinda like gyro chicken you'll find at a restaurant. You can leave them whole, but I like to put out a platter of the chopped chicken, tomatoes, cukes, olives, hummus, pita, tzatziki, feta, etc., and let people cobble together a light meal.