For the shawarma: Mix the cumin, paprika, allspice, chili powder and garlic together. Place the chicken inside a resealable plastic bag with the olive oil. Add the spice mixture and rub in until fully incorporated. Seal that bag and place it into the refrigerator to marinate for at least 30 minutes.
Preheat the grill to medium-high heat.
After marinating, use the skewers to pierce the pieces of chicken through the center. Slide the pieces down leaving enough room for you to hold the skewer, about 3 pieces of chicken per skewer. Continue this process until chicken is all skewered. Sprinkle with salt and place on a hot oiled grill top for 4 minutes per side, or 8 minutes total.
For the relish: Mix the oil, vinegar, oregano, tomatoes, cucumbers, garlic, lemon juice, onion, parsley and some salt and pepper.
For the tahini sauce: In a medium bowl, combine the tahini paste, olive oil, lemon juice and 1/4 cup water. Season with salt.
To serve, fill the pitas with the grilled chicken and serve with the cucumber relish and tahini sauce.