To make the teriyaki sauce, heat the rice wine vinegar, brown sugar, soy sauce, ginger, and scallions to boiling over medium heat. Let cook for 7 minutes, then whisk in cornstarch slurry and cook for another 3 minutes or so, to allow to thicken. Remove from heat and cool to room temperature.
Arrange the chicken cubes in a shallow nonreactive container such as a glass baking dish. Season with salt and pepper and rub into the chicken. Pour the sake over the chicken and then pour in the teriyaki sauce. Cover and let marinate in the refrigerator for at least 2 hours.
Begin soaking about 12 bamboo skewers in water so they won't burn on the grill.
Heat the grill and thread alternate pieces of chicken and leeks on the skewers. Each skewer should have approximately 3 pieces of chicken, and each serving will consist of 3 skewers. Grill over indirect heat until the chicken is tender but not dry, about 5 to 10 minutes.