In large bowl whisk together all the souvlaki ingredients, except the chicken. Reserve 1 1/2 tablespoons for Round 2 Recipe Greek Flat Bread. Add the chicken and toss to coat. Cover and let marinate for at least 30 minutes or up to overnight in the refrigerator.
Heat a large skillet over medium-high heat. Remove the chicken from the marinade and add to the pan. Cook until the chicken is cooked through, about 10 to 12 minutes.
In a medium bowl combine all the tzatziki sauce ingredients and set aside.
Divide the dough into 4 equal portions. On a floured work surface, roll out each portion into 6 to 7 inches in diameter and about 1/8-inch thick.
Preheat a grill pan or grill over medium heat.
Grill the dough for 2 to 3 minutes per side. The pitas should be cooked through but still pliable and not too crispy.
Put a grilled pita onto a plate, top with some cooked chicken, lettuce, tomato, onion and some tzatziki sauce. Repeat with remaining ingredients and arrange on a serving platter.
Reserve the leftover tzatziki sauce, chicken, tomatoes and onion for Greek Flat Bread.