1. Combine yogurt, olive oil, lemon juice, garlic, dill, and pepper in a bowl. Cover and refrigerate for one hour.
2. Preheat grill to medium high or position rack in upper third of oven and preheat broiler.
3. Sprinkle chicken with seasoning. Place the chicken on the grill rack or broiler pan and cook until no longer pink in the middle and an instant-read thermometer inserted into the thickest part reads 165°F, 4 to 8 minutes per side, depending on the size of the breast. Transfer the chicken to a clean cutting board and let cool.
4. Add cucumber to yogurt mixture. Salt to taste.
5. Thinly slice the chicken. Split open the warm pitas and fill with chicken, tzatziki sauce, lettuce, tomato, and onion.