Prepare the chicken: Remove any residual skin or fat from the chicken; pat dry and transfer the chicken pieces to a bowl and set aside.
Prepare the marinade: In a mixing bowl, whisk together the yogurt, lemon zest, lemon juice, olive oil, garlic, ground cumin, dried oregano, paprika, tomato paste, salt, and pepper so that all are incorporated.
Pour the marinade over the chicken and mix well, ensuring all the chicken pieces are thoroughly coated. Cover it tightly with plastic wrap, removing as much air as possible. Place it in the fridge and marinate for at least 4 hours or preferably overnight.
If using wooden skewers, begin soaking them in water for 30 minutes prior to assembling the marinated chicken. Remove the marinating chicken from the fridge 20 minutes before grilling.
When ready to grill, turn burners to medium-high and let them preheat for 10 minutes or until it reaches 400 degrees F. Using tongs, dip a wad of paper towel into the extra olive oil and then generously wipe the grates.
Meanwhile, remove the chicken from the marinade and thread the chicken pieces through the skewer. Tap off any excess marinade. If you have a thin piece, fold it as you thread it so it will cook evenly. Do not pack the pieces too tight, as this can lead to uneven cooking.
Place the skewers on the oiled grate in a single layer. Cover and cook for approximately 12-15 minutes, rotating every 5 minutes so that you achieve a nice browning with grill marks. If you have a meat thermometer, check doneness by inserting it into one of the chicken skewers. If the internal temperature reached to 165 degrees F., your shish tawook should be cooked.
Remove the skewered chicken from the grill and tent with aluminum foil for at least 10 minutes before serving.