Scanning tunneling microscopy experiments show that the unstable growth morphology observed during molecular beam homoepitaxy on slightly vicinal Si(001) surfaces consists of straight step bunches. The instability occurs under step-flow growth conditions and vanishes both during low-temperature island growth and at high temperatures. An instability with the same characteristics is observed in a 2D Kinetic Monte Carlo model of growth with incorporated Si(001)-like diffusion anisotropy. This provides strong evidence that the diffusion anisotropy destabilizes growth on Si(001) and similar surfaces towards step bunching. This new instability mechanism is operational without any additional step edge barriers.