A novel terahertz (THz) Fabry-Perot cavity (FPC) antenna is proposed, based on a multistack of alternating layers of highly birefringent nematic liquid crystal (NLC) and high-permittivity dielectric, which comprehensively acts as a Bragg reflector. This layout is able to provide enhanced reconfigurability and improved directivity with respect to standard FPC leaky-wave antenna (LWA) designs. The application of a low driving voltage to the NLC layers allows for the dynamic control of the radiating properties of the LWA, at a fixed frequency in the low-THz range. The attractive features and performance of two different configurations are rigorously discussed in a theoretical context, taking into account the realistic implementation of the proposed device.