Combine 1/3 cup yogurt and next 5 ingredients (yogurt through red pepper) in a small bowl. Thread chicken pieces onto four (10-inch) skewers. Brush chicken with yogurt mixture. Chill 10 minutes.
Place skewers on a grill rack coated with cooking spray; cook 4 minutes on each side or until done. Remove chicken from skewers.
Combine the remaining yogurt, cucumber, cilantro, onion, and sugar in a small bowl. Combine chicken, 1/4 cup cucumber mixture, and greens in a large bowl, tossing gently to coat. Fill each pita half with 1/2 cup chicken mixture. Serve with remaining cucumber mixture.