Preheat broiler to medium. Place rack 4 inches from heat source. Line a rimmed baking sheet with foil. Combine oil, garlic, lemon zest and 1 tsp. salt in a large ziplock bag. Add chicken, seal, and toss to coat. Set aside while broiler heats. In a medium bowl, mix cucumbers, yogurt and mint; season with salt and pepper.
Thread chicken on 4 metal skewers. Place skewers on baking sheet and broil, turning once, until chicken is cooked through, 12 to 15 minutes total.
Serve chicken skewers with cucumber mixture, cherry tomatoes and pitas on the side.