To make the marinade, mix together the soy sauce, sesame oil, minced garlic, and pepper flakes in a small bowl. Place the chicken strips together with the marinade in a zip lock bag and toss to coat. Leave in a fridge for at least an hour.
Transfer the chicken to a large skillet, discarding the marinade. Cook for 12-15 minutes, until cooked through. When done, transfer to a plate and let cool enough to handle.
Soften the rice paper rounds by submerging them for a few seconds in a bowl with warm water. Some packages recommend to dip wrappers for about a minute but I wouldn't recommend doing so because they will become too soft. Work with one rice paper round at a time.
Place some cabbage, bell pepper, radish, carrot, cilantro, and chicken on each round (about 1 1/2 inch from the bottom). Fold the sides and then the bottom, rolling it up. Repeat with all the remaining ingredients. Serve with the dipping sauce.