For the tzatziki:
For the chicken:
To make the tzatziki sauce, strain the yogurt using cheesecloth over a bowl. Let strain for several hours or overnight, if possible, to remove as much moisture as possible.
Shred the cucumber. Wrap in a towel a squeeze to remove as much water as possible. Mix together the strained yogurt, shredded cucumber, garlic, white wine vinegar, salt and pepper to taste, and lemon juice. Drizzle lightly with olive oil. Refrigerate for at least 30 minutes before serving to allow the flavors to meld.
To prepare the chicken, combine the garlic, lemon juice, red wine vinegar, olive oil, yogurt, oregano, and salt and pepper to taste in a medium bowl. Whisk together until mixed well. Add the chicken pieces to the bowl and mix well to coat. Cover and refrigerate for about 1 hour.
Cook the chicken as desired, either in the skillet or with the broiler. (I butterflied the chicken breasts and then cooked them under the broiler.) Once the chicken is completely cooked through, transfer to a plate and let rest for 5 minutes. Cut into strips.
Heat pitas (I sprayed with some spray oil and then popped them in the toaster oven for a few minutes). Top with chicken, tzatziki sauce, diced tomatoes and sliced onions. Serve immediately.