This paper presents an efficient method to compute the dispersion diagram of periodic structures with generic anisotropic media. The method takes advantage of the ability of full-wave commercial simulators to deal with finite structures having anisotropic media. In particular, the proposed method opens new possibilities with respect to commercial eigenmode solvers: (i) anisotropic materials with non-diagonal permittivity and permeability tensors can be analyzed; (ii) the attenuation constant can easily be computed and lossy materials can be included in the simulation; (iii) unbounded and radiating structures such as leakywave antennas can be treated. In this work, the proposed method is particularized for the study of liquid crystals (LCs) in microwave and antenna devices. Since LCs show promising capabilities for the design of electronically reconfigurable elements, the dispersion properties of a great variety of LC-based configurations are analyzed, from canonical structures, such as waveguide and microstrip, to complex reconfigurable phase shifters in ridge gap-waveguide technology and leaky-wave antennas. Our results have been validated with previously reported works in the literature and with the Eigenmode solver of commercial software CST.