Combine feta cheese, yogurt, dill, 1 teaspoon oil, and 1/4 teaspoon garlic in a small bowl, stirring well.
Heat remaining 2 teaspoons olive oil in a large skillet over medium-high heat. Add remaining 1 teaspoon garlic and oregano to pan, and sauté for 20 seconds. Add chicken, and cook for 2 minutes or until thoroughly heated. Place 1/4 cup chicken mixture in each pita half, and top with 2 tablespoons yogurt mixture, 2 tablespoons shredded lettuce, 1 tablespoon cucumber, and 1 tablespoon tomato. Divide onion evenly among pitas.