In a medium bowl, whisk together 2 tablespoons olive oil, oregano, garlic powder, salt, and black pepper. Toss chicken strips in the marinade to coat.
Heat a skillet or grill pan over medium-high heat. Cook the chicken strips for 5 to 7 minutes per side or until the internal temperature reaches 165°F (74°C). Remove from heat.
In a small bowl, prepare the tzatziki sauce by combining the Greek yogurt, grated cucumber, lemon juice, minced garlic, and fresh dill. Stir until smooth.
Lightly warm the pita breads in a dry skillet or microwave for 15 seconds.
To assemble, spread 2 tablespoons of tzatziki sauce in the center of each pita.
Divide the shredded lettuce, diced tomato, sliced cucumber, and red onion evenly among the pitas.
Place the grilled chicken strips on top of the vegetables.
Garnish with crumbled feta cheese, fold the pita over the filling, and serve immediately.