This paper presents an analysis using computational and scaling methods of pulsatile capillary Poiseuille flow of a model anisotropic viscoelastic discotic liquid crystalline material. The analysis shows that pulsatile pressure drops applied to anisotropic materials can result in flow-rate enhancement or reduction, defined as the relative flow-rate change with respect to the steady-state flow rate, for a given average pressure drop, amplitude, and frequency. It is found that flow-rate modification in pulsatile flow is a direct result of orientation-dependent viscosity. The role of average pressure drop, oscillation amplitude, and frequency on flow enhancement is characterized and explained. The new mechanism of flow-enhancement in liquid crystals subjected to pulsatile pressure is expected to be useful to the fundamental understanding of pulsatile flows of anisotropic suspensions and anisotropic biological fluids.