In a large bowl, combine the chicken pieces with olive oil, lemon juice, minced garlic, cumin, paprika, turmeric, coriander, allspice, cinnamon, salt, and pepper. Make sure every piece of chicken is nicely coated. Cover and marinate in the refrigerator for at least 30 minutes (or longer for more flavor).
While chicken marinates, whisk together tahini sauce ingredients: tahini, lemon juice, water, minced garlic, and salt in a small bowl. Adjust water for desired consistency and salt to taste.
Heat a large skillet or grill pan over medium-high heat. Add marinated chicken in a single layer and cook for 5-7 minutes per side, until fully cooked (165°F/74°C internal temperature) and browned.
Warm flatbreads in a skillet, microwave, or oven until pliable.
Assemble wraps: Lay warm flatbread, add cooked chicken, chopped red onion, tomatoes, cucumber, and parsley. Drizzle generously with tahini sauce.
Fold in sides of flatbread, then tightly roll up from the bottom. Slice in half and serve immediately.