1. Rinse chicken; pat dry with paper towels. Sprinkle chicken lightly with salt and pepper.
2. In a covered grill arrange medium-hot coals around drip pan. Test for medium heat above the pan.
3. Place chicken, skin side up, on the grill rack over the drip pan but not over the coals. Lower grill hood. Grill chicken for 50 to 60 minutes or until tender and no longer pink, adding more coals as needed.
4. Meanwhile, combine salsa and lime juice. Set aside 1/2 cup of the mixture. Brush chicken occasionally with the remaining salsa mixture during the last 10 to 15 minutes of grilling.
5. For sauce, in a small saucepan combine the reserved salsa mixture, the cucumber, green sweet pepper, and cilantro. Bring just to boiling. Remove from heat. Serve the cucumber mixture with chicken. Makes 4 servings.
* If you make this recipe with chicken quarters, break the wing, hip, and drumstick joint so the quarters will lie flat on the grill. Also, twist the wing tips under the backs.