To cut a workpiece along two different cutting directions, the band saw teeth are set horizontally by a band saw guide device and then the cutting head (CH) is moved frontward, for instance; thereafter, the band saw teeth are twisted downward by the same guide device and then the cutting head is moved downward. The machine is effectively usable in cutting off a test piece from a large workpiece. To improve the cutting precession, various novel devices are incorporated, such as a balance cylinder (B-CYL) for allowing the cutting head to move up and down smoothly; two back-and-forth cylinders (BF-CYL) for moving the cutting head back and forth in symmetrical pressure application; cutting head fix device (FD); a rigid guide column; band saw holding device (BSH); a slidable small-diameter driven wheel arranged within the cutting head (CH), etc.