In large bowl, whisk together yogurt, oregano, oil, lemon rind and juice, garlic, salt and pepper.
Add chicken; turn to coat.
Cover and refrigerate for 4 hours, turning occasionally. (Make-ahead: Refrigerate for up to 24 hours)
Place chicken on greased grill over medium heat; brush with any remaining marinade.
Close lid and grill, turning once, unitl golden brown, grill-marked and no longer pink inside for breasts or until juices run clear for thighs, about 12 minutes.