For marinade: Combine yogurt, lemon juice, garlic powder, oregano, and red wine vinegar with chicken in gallon-size zip-top bag. Coat chicken evenly and marinate 20 minutes or overnight.
For cucumber salad: In large bowl, combine cucumbers, tomatoes, red onions, ¼ cup olive oil, lemon juice, red wine vinegar, oregano, and garlic powder; set aside.
In large skillet over medium-high heat, add 1 tablespoon olive oil. Remove chicken from marinade and cook 3-4 minutes per side.
Evenly divide chicken, cucumber salad, tzatziki, and cheese into serving bowls.
Note: Extended marinating time required.