Preheat the oven to 400 degrees Fahrenheit (200 degrees Celsius).
On a large rimmed sheet pan, combine the chicken strips, bell peppers, and red onion.
Drizzle with olive oil and sprinkle with oregano, garlic powder, salt, and pepper; toss thoroughly to coat.
Arrange the mixture in a single, even layer across the sheet pan.
Roast for 18 to 20 minutes, or until the chicken reaches an internal temperature of 165 degrees Fahrenheit (74 degrees Celsius).
While the chicken roasts, whisk together the Greek yogurt, mayonnaise, dill, chives, lemon juice, and minced garlic in a small bowl to create the ranch dressing.
Place the pita breads directly on the oven rack for the final 2 minutes of cooking time to warm through.
Open the warm pitas and fill with the roasted chicken and vegetable mixture.
Drizzle generously with the prepared herby ranch dressing and serve immediately.