Place 1 chicken breast half, boned side up, between two pieces of plastic wrap or waxed paper.
Working from center, gently pound chicken with rolling pin or flat side of meat mallet until about 1/4" thick; repeat with remaining chicken pieces.
In 12x8" baking dish or plastic bag, combine all marinade ingredients; mix well to dissolve sugar.
Add chicken; turn to coat.
Cover dish or seal bag; refrigerate 1 to 2 hours, turning several times.
Drain chicken, reserving marinade.
When ready to barbecue, place chicken on gas grill over low heat or on charcoal grill 4 to 6 minches from medium coals.
Cook 10 to 15 minutes or until chicken is no longer pink, turning once and brushing with reserved marinade.
Discard any remaining marinade.