Soak bamboo skewers in water at least 30 minutes before using to prevent burning. In medium bowl, mix chicken and 1/4 cup teriyaki marinade. Cover; refrigerate 30 to 60 minutes to marinate.
Meanwhile, heat coals or gas grill for direct heat. In small bowl, stir all sauce ingredients until blended; set aside.
Remove chicken from marinade; discard marinade. Thread chicken on skewers. Place skewers on grill. Cover grill; cook over medium heat 8 to 10 minutes, turning once, until chicken is no longer pink in center.
Meanwhile, cook rice in water as directed on package, omitting butter and salt. Sprinkle chicken with additional chopped fresh cilantro if desired. Serve with sauce and rice.