Heat grill to high (450° to 550°). Mix chicken, oil, wine, lemon juice, oregano, crushed garlic cloves, salt, and 1/4 tsp. pepper in a bowl. Thread chicken onto 8 metal skewers.
Grill chicken, turning occasionally, until grill marks appear and meat is cooked through, 10 to 12 minutes.
Put cucumber in a kitchen towel and wring out any excess liquid. Transfer to a medium bowl. Add yogurt, minced garlic, remaining 1/4 tsp. pepper, and the vinegar and mix until combined. Serve with chicken and warm pita bread.