In a bowl, mix yogurt, green onions, mint, cumin, salt and 1/8 teaspoon of pepper. Pour half the sauce in a bowl and set aside. Add cucumber to the remaining sauce and stir. Set aside.
Sprinkle the chicken breasts with remaining pepper. Set the barbecue to medium heat. Put the chicken breasts on an oiled grill of the barbecue and cook, without closing the cover, 12 to 15 minutes or until it's not pink inside. (flip them and brush with the sauce without cucumber at mid-cooking; discard remaining of this sauce). Serve with the sauce with cucumber.