Orbital angular momentum (OAM) has been extensively studied to date and has become topical in the last few years due to its potential for increasing bandwidth in optical communications. The so-called Dammann vortex grating (DVG) can generate a series of equal-energy OAM beams and satisfactorily realize OAM parallel detection, an important challenge in this field. However, previously reported DVGs suffer from polarization sensitivity, low efficiency, or the lack of tunability and mode variety. Here, a design of liquid-crystal DVGs is proposed and demonstrated for the generation of various equal-energy OAM beams. The DVGs, featured by alternative orthogonally planar-aligned regions, are carried out via photopatterning technology. Beam arrays composed of arbitrary OAM modes, as well as two-dimensional ones, are generated in good quality and high efficiency. The liquid-crystal DVGs exhibit merits of excellent polarization independency, electrical switchability, and tunability. This supplies a promising approach towards OAM generation, manipulation, and detection.