Stir yogurt, 2 tablespoons dill, garlic and 1 teaspoon lemon juice in small bowl to blend. Season with salt and pepper.
Place chicken in medium bowl. Sprinkle with oregano, 1 teaspoon dill and salt and pepper. Toss to coat.
Heat 1 tablespoon oil in heavy large skillet over medium-high heat. Add chicken; saute until brown and cooked through, about 5 minutes. Transfer to plate. Add 1 tablespoon oil to skillet. Add onions; saute until beginning to brown, about 10 minutes. Return chicken and any juices to skillet. Add 1 tablespoon lemon juice. Stir until heated through, about 2 minutes.
Arrange pita rounds on plates. Top pita rounds with chicken mixture. Spoon some yogurt sauce over chicken. Serve, passing extra sauce separately.