Whisk soy sauce, sake, sugar, and ginger together in a bowl. Add chicken cubes; toss to coat. Marinate in the refrigerator for 2 to 3 hours.
Thread chicken cubes and green onions onto skewers.
Pour marinade into a small saucepan over medium heat; bring to a simmer. Reduce heat to low and keep warm.
Preheat an outdoor grill for medium-high heat and lightly oil the grate. Grill skewers until chicken is browned, for 3 minutes. Flip and brush with warm marinade. Cook until second side is browned, about 3 minutes more.