In a bowl, whisk together 2 tablespoons olive oil, 1 tablespoon lemon juice, oregano, garlic powder, salt, and pepper. Add the chicken breasts and toss to coat. Marinate for at least 30 minutes.
In a separate bowl, combine Greek yogurt, remaining olive oil and lemon juice, minced garlic, dill, mint, salt, and pepper. Whisk until smooth and creamy.
Slice the cucumber, tomatoes, and red onion.
Heat a grill pan or skillet over medium-high heat. Remove the chicken from the marinade and grill for 5-7 minutes per side, or until cooked through.
Let the chicken rest for 5 minutes before slicing it into thin strips.
Warm the pita bread in a dry skillet or in the microwave for a few seconds until soft and pliable.
Spread a generous spoonful of the Lemon Herb Sauce on each pita bread.
Add the sliced chicken, cucumber, tomatoes, and red onion.
Drizzle with more Lemon Herb Sauce, if desired.
Fold the pita bread in half or roll it up, making sure ingredients are equally distributed. Serve immediately.