In a large bowl or zip-top bag, combine the Greek yogurt, olive oil, lemon juice, lemon zest, garlic, thyme, paprika, cumin, oregano, cayenne pepper, salt, and black pepper. Stir until the mixture is smooth and evenly blended. The yogurt acts as the base that carries all the seasonings onto the shrimp.
Add the shrimp to the marinade and toss until every piece is coated. Cover the bowl or seal the bag and refrigerate for about 30 minutes. Avoid marinating for more than 2 hours because the lemon juice can start affecting the shrimp's texture.
If using wooden skewers, soak them in water before grilling to prevent burning.
Heat your grill to medium-high heat and lightly oil the grates. A properly heated grill helps create attractive grill marks and prevents sticking. Make sure the cooking surface is clean before placing the skewers on it.
Place the shrimp skewers on the hot grill and cook for about 3 minutes per side. The shrimp should turn pink on the outside and opaque in the center. Be careful not to overcook them, as shrimp can become rubbery very quickly.
Transfer the cooked skewers to a serving platter. Sprinkle with fresh parsley and crumbled feta cheese, then add lemon wedges on the side. The fresh garnishes bring extra color and Mediterranean flavor to the finished dish.