Grate cucumber and squeeze dry using a towel.
Mix yogurt, garlic, dill, lemon juice, olive oil, salt, and pepper to make tzatziki.
Chill tzatziki in refrigerator for 30 minutes.
Season and grill chicken for 6–7 minutes per side until cooked through (165°F). Rest for 5 minutes.
Slice vegetables evenly for sandwich assembly.
Warm pita breads on skillet or grill pan for 30 seconds per side.
Spread tzatziki on pita, layer chicken, vegetables, feta, and olives.
Fold pita and cut sandwich in half. Serve immediately.