[math-fun] Path Question

Suppose there is a man standing on a random square of a random polyomino. At regular intervals, we may instruct the man to move in one of the directions north, south, east or west. If the adjacent square in the specified direction is part of the polyomino, the man will move to that square, otherwise he will stay on his current square. Is there a single infinite sequence of directions we can give to the man that guarantees he will eventually visit every square of the polyomino, regardless of its shape or size or of the man's initial square?