The tunable optical diffractive properties of zenithal bistable nematic liquid-crystal gratings are theoretically investigated. The liquid-crystal orientation is rigorously solved via a tensorial formulation of the Landau-de Gennes theory and the optical transmission properties of the gratings are investigated via full-wave finite-element frequency-domain simulations. It is demonstrated that by proper design the two stable states of the grating can provide nondiffracting and diffracting operation, the latter with equal power splitting among different diffraction orders. An electro-optic switching mechanism, based on dual-frequency nematic materials, and its temporal dynamics are further discussed. Such gratings provide a solution towards tunable beam-steering and beam-splitting components with extremely low power consumption.