In a large Tupperware container, mixing bowl, or Ziplock bag, combine steak cubes and sweet and spicy soy marinade. Marinate for at least 2 hours or up to overnight, covered in the refrigerator.
When almost ready to grill, soak 6 wooden skewers in water for at least 30 minutes. Preheat a clean grill over medium heat. If using a grill pan, wait to preheat until ready to cook.
Add the bell peppers and mushrooms to the marinade and toss to coat completely (You may need to remove steak from marinade to make room for the veggies).
Prepare the skewers in this order: bell pepper, shiitake, steak, bell pepper, shiitake, steak, bell pepper, shiitake (2 pieces of steak, 3 bell peppers, and 3 shiitakes per skewer). Reserve marinade for basting.
Right before grilling, sprinkle skewers with Kosher salt and freshly ground black pepper.
Cook the skewers for 6-9 minutes total, turning as each side caramelizes and basting with the reserved marinade, until cooked to preference and veggies tender.
Serve sprinkled with thinly sliced scallions and toasted sesame seeds.