A band saw machine which comprises a generally elongated frame structure including an endless band saw trained between and adapted to travel over a pair of spaced guide wheels and having cutting and return reaches therebetween, a positioning mechanism for fixedly supporting a stock to be cut, and a rocking mechanism including a pair of spaced reciprocating drives for rocking the endless band saw through a predetermined angle in the opposite directions generally about the stock to be cut, with the opposite ends of the frame structure alternately upwardly and downwardly shifted. The frame structure is supported for movement relative to the positioning mechanism between inoperative and cutting positions in a direction generally perpendicular to the stock. The rocking mechanism is operable at least during the cutting operation performed by the cutting reach of the endless band saw.