In a medium bowl, combine yogurt, green onions, mint, cumin, salt, and 1/8 teaspoon pepper. Transfer half of the yogurt mixture to a small bowl; set aside. For cucumber-yogurt sauce: Stir cucumber into remaining yogurt mixture.
Sprinkle chicken breasts with 1/8 teaspoon pepper.
Place chicken on the rack of an uncovered grill directly over medium coals. Grill for 12 to 15 minutes or until chicken is no longer pink (170 degrees F), turning once halfway through grilling and brushing with reserved yogurt mixture for the last half of grilling. Discard any remaining yogurt mixture.