Combine teriyaki sauce and oil in a shallow dish or large zip-top plastic freezer bag; add steaks, turning to coat. Cover or seal, and chill 1 hour, turning steaks occasionally.
Remove steaks from marinade, discarding marinade.
Grill steaks, covered with grill lid, over medium-high heat (350° to 400°) 5 to 7 minutes on each side or to desired degree of doneness. Let stand 5 minutes; cut diagonally across the grain into thin slices.
Toss steak slices, cabbage, and next 5 ingredients together; drizzle with desired amount of dressing, tossing gently to coat.