We report the terahertz (THz) emission from the wurzite indium nitride (InN) films grown by molecular beam epitaxy (MBE). More than two orders of magnitude of THz power enhancement has been achieved from the InN film grown along the a-axis and magnesium (Mg) doped InN with a critical carrier concentration. The primary radiation mechanism of the a-plane InN film is found to be due to the acceleration of photoexcited carriers under the polarization-induced in-plane electric field perpendicular to the a-axis. Apparent azimuthal angle dependences of THz wave amplitude and the second harmonic generation are observed from a-plane InN. In the Mg-doped films, Mg as the acceptors compensate the native donors in the InN films and large band bending over a wider space-charge region causes the enhancement of THz emission power compared to the undoped InN.