**Marinate the Chicken:** Combine chicken, yogurt, lemon juice, olive oil, vinegar, salt, pepper, and oregano in a bag. Let it marinate for at least 30 minutes, preferably 2-4 hours, or ideally overnight.
**Prepare Tzatziki Sauce:** Mix yogurt, olive oil, lemon juice, vinegar, salt, dill, and garlic. Add grated cucumber after squeezing out excess water. Adjust seasonings to taste.
**Make Greek Salad:** Toss tomatoes, cucumber, red onion, parsley, olive oil, salt, and pepper in a bowl. Season to taste.
**Cook the Chicken:** Grill chicken on kebobs for 5-7 minutes per side or cook in a skillet for 8-10 minutes until fully cooked and golden brown.
**Assemble the Gyros:** Place chicken in pita bread, warm it up if desired. Top with tzatziki sauce and Greek salad. Add extra feta if you'd like.