In a small bowl, combine the ingredients for the marinade (yogurt, lemon juice, garlic, cinnamon, oregano, salt, nutmeg, and clove). Add chicken to the marinade, coat well, cover and refrigerate for 4-24 hours. If desired, slice the chicken into strips before marinating to maximize the flavor.
Prepare the yogurt sauce for the sandwiches. Open a 7 oz. container of plain greek yogurt and stir in the garlic, dill, and salt. Refrigerate until ready to use.
After marinating the chicken, grill until cooked through (I use a George Foreman, cooked for about 7 minutes).
Prepare the vegetables for the sandwich. Chop, rinse, and drain the lettuce in a colander. Wash and slice the cucumber and tomato.
Build the chicken shawarma sandwiches. Spread the yogurt sauce on your flat bread, top with the chicken and vegetables. Roll the sandwich closed. You can wrap the sandwich in foil to help it stay closed as you eat.