Recently, the terahertz (THz) quantum cascade lasers and intersubband transition devices has attracted considerable attention based on AlGaN/GaN quantum well structure for far-infrared applications. It is necessary to investigate the optical and dielectric properties of GaN and sapphire crystals in the far IR as well as THz frequency regions. In the previous report, nonpolar and semipolar GaN crystals in the hexagonal crystal system are found to show multi-phonon mode behavior in the far-infrared frequency region. In addition, the polarized IR reflectance response of sapphire is reported to exhibit the dependence on the crystal orientation. In this paper, we investigated the optical properties and dielectric response of nonpolar a-plane GaN (a-GaN) films in the terahertz (THz) frequency using polarized femto-second THz TDS. The phase separation and amplitude ratio were measured by changing the crystal orientation with respect to the polarization of the incident THz pulse. In sharp contrast with polar c-plane GaN, nonpolar a-GaN films exhibited the angle dependence of the THz absorptions at room temperature, presumably due to the birefringence in nonpolar a-GaN films. The transmitted THz signal intensity, the time delay, and the Fourier-transformed amplitude were investigated by rotating the azimuth angle of the crystal orientation. Strong THz field absorptions around 1 THz frequency were observed at the azimuth angle of 0°, 90°, and 180° for a-GaN films. We believe that the birefringence in the THz frequency region originates from the difference of the transverse-optical and the longitudinal optical phonon splitting between the optical phonon branches.