Cut the steak crosswise into 2 to 4 pieces and place in a bowl. Pour teriyaki sauce over the steak, turning to coat well. Set aside.
In a medium skillet, heat oil over medium-high heat. Chop onion. Add onions and mushrooms, and saute for 8 to 10 minutes or until softened.
Meanwhile, heat a lightly greased ridged grill pan over moderately high heat until just smoking.
Remove steak from marinade and grill 3 minutes on each side for medium-rare meat. (Alternatively, broil steak on rack of a broiler pan about 4 inches from heat.) Transfer steak to a platter and let stand, uncovered, 5 minutes.
With a sharp knife diagonally cut steak into 1/3-inch-thick slices and serve with vegetables over hot rice, if desired.