Cut white and pale green parts of green onions into 1 1/2" pieces. Thinly slice dark green onion tops. Set aside.
Combine marinade, ginger, garlic, sugar, and sesame oil in a zip-top plastic freezer bag; add chicken and 1 1/2" green onion pieces, turning to coat. Seal and chill for at least 30 minutes or up to 2 hours.
Meanwhile, soak 8 (6") wooden skewers in water to cover for 30 minutes. Drain.
Thread marinated chicken and green onion pieces onto skewers, discarding marinade. Grill over medium-high heat (350° to 400°) for 3 to 4 minutes on each side or until chicken is done.
Transfer skewers to an appetizer platter, and sprinkle with thinly sliced green onions.