In a large bowl, mix 3 tablespoons olive oil, 2 tablespoons lemon juice, 3 minced garlic cloves, 2 teaspoons oregano, salt, and pepper. Add the chicken cubes and toss to coat. Let the chicken marinate for 15 minutes while you prepare the rest of the meal.
In a medium bowl, stir together the Greek yogurt, grated cucumber, remaining minced garlic clove, 1 tablespoon lemon juice, and dill to make the tzatziki. Cover and refrigerate until serving.
In another bowl, combine the cherry tomatoes, diced cucumber, red onion, feta, and olives. Drizzle with the remaining 2 tablespoons olive oil and sprinkle with the remaining 1 teaspoon oregano. Toss gently and set aside.
Thread the marinated chicken onto skewers. Heat a grill or grill pan over medium-high heat and cook the skewers for 10 to 12 minutes, turning once, until the chicken is cooked through and lightly charred.
Warm the pita breads on the grill for 1 minute per side. Remove the chicken from the skewers and serve everything together so each person can assemble their own plate.