In a gallon size zip top bag combine the garlic cloves, lemon juice, vinegar, olive oil, oregano, dill, salt, and pepper. Add the chicken to the bag and marinate in the refrigerator for at least 4 hours.
Cook the chicken on the grill until it reaches 170 degrees internally, about 5-7 minutes per side. Remove to a cutting board and allow to rest for 5 minutes.
Cut the chicken into thin slices. Layer the chicken, red onions, lettuce, feta, and tzatziki sauce on the pita bread.
Roll them up and serve with additional tzatziki.