Mix the ground turkey, oregano, cumin, salt and pepper and form into patties. I like to do this ahead and place the patties on waxed paper on a plate and store them in the fridge until dinner.
Mix together the olives, yogurt, lemon juice and more black pepper and set aside. You can do this ahead of time too--just cover the bowl and store in the fridge.
When you're ready to cook, heat a cast iron skillet or your outdoor grill and cook the patties until they're done to your liking, 4-5 minutes per side depending on how hot your grill/pan is.
Toast the buns.
Spread the sauce on both sides of the buns and top with as much arugula as you like. I liked a lot. Put the patty on top of the argula, then top with some sliced red onion and cucumber. Put on the top of the bun and serve!