In a large mixing bowl, combine minced garlic with ground cumin, coriander, paprika, turmeric, cayenne pepper, cinnamon, olive oil, salt, and black pepper. Mix well and add chicken strips, tossing to coat. Cover and marinate in the refrigerator for at least 1 hour or overnight.
Preheat a grill pan or non-stick skillet over medium-high heat. Cook marinated chicken strips in batches for 6-8 minutes per side until cooked through and slightly charred. Remove from heat and let rest before slicing into bite-sized pieces.
Prepare fresh vegetables by dicing cucumber, halving cherry tomatoes, and slicing red onion. Place in a bowl for easy access.
Warm pita or flatbreads in a dry skillet over medium heat for about 1 minute on each side. Lay one flatbread on a plate, add a portion of cooked chicken, and layer with cucumber, cherry tomatoes, and red onion.
Drizzle plain yogurt or tahini sauce over the filling for added creaminess.
Fold the sides of the pita over the filling and roll tightly from the bottom to create a secure wrap. Serve immediately, garnished with fresh parsley and lemon wedges.