Place meat between two pieces of plastic wrap and pound out with the flat side of a meat mallet. Remove plastic wrap and score steak on both sides by making shallow cuts at 1-inch intervals. Using half of the teriyaki sauce, marinade the steak for a few hours in the refrigerator.
In a medium bowl, combine spinach, water chestnuts, green onions, and 2 tablespoons of the teriyaki sauce.
Spread spinach mixture over steak. Starting from a short side, roll steak up. Secure with wooden toothpicks at 1-inch intervals, starting 1/2 inch from one end. Slice between toothpicks into eight 1-inch thick slices. Thread two slices onto each four long metal skewers. (If using wooden skewers, make sure to soak them in water for at least 30 minutes.).
Heat a grill to medium. Places slices on grill rack, cover, and cook for 7 minutes per side, brushing with some teriyaki sauce halfway through.