In a medium bowl, combine the olive oil, garlic, lemon juice, cumin, paprika, turmeric, coriander, cinnamon, allspice, cayenne pepper (if using), salt, and pepper.
Add the thinly sliced chicken to the marinade and stir well to coat. Cover the bowl and refrigerate for at least 30 minutes, or preferably overnight, to allow the flavors to deepen.
Preheat the grill or grill pan over medium-high heat.
Once the grill is hot, place the marinated chicken slices onto the grill. Cook for 4-5 minutes per side, or until the chicken is cooked through and has nice grill marks.
While the chicken is grilling, warm the pita or flatbreads on the grill for about 1-2 minutes, just until they are soft and pliable.
Remove the chicken from the heat and slice it into strips if needed.
To assemble, place a few pieces of chicken in the center of each warm pita, drizzle with tahini or garlic yogurt sauce, and add optional toppings like cucumber, tomato, and red onion. Garnish with fresh parsley.
Roll up the pita tightly, folding in the sides, and serve immediately.