In a bowl, combine soy sauce, sake or white wine, mirin, sugar, vegetable oil, salt, and black pepper.
Cut the chicken breasts into strips and add them to the bowl with the marinade. Toss until the chicken is well coated. Let it marinate for at least 10 minutes.
Preheat a grill or grill pan over medium-high heat.
Remove the chicken from the marinade, reserving the marinade for basting.
Thread the chicken onto skewers, alternating with the green onions.
Grill the chicken skewers for about 3-4 minutes per side, basting with the reserved marinade.
Remove the chicken from the grill and let it rest for a few minutes.
Warm the tortillas or lettuce leaves.
Place a few pieces of chicken and green onions onto each tortilla or lettuce leaf.
Roll up the tortilla or wrap the lettuce leaf, securing with toothpicks if necessary.
Serve the Yakitori-Style Chicken and Green Onion Wraps with extra marinade for dipping.