A theory of electrically controllable long-period gratings built in a liquid-crystal fiber has been presented by means of an anisotropic waveguide analysis and the coupled-mode theory with the discretization method. Effective propagation and coupling constants have been found for a circular cross-sectional fiber with a nematic liquid-crystal core, the optic axis of which is in the direction of the fiber axis. The results show that when uniaxial perturbations of the liquid-crystal director exist in the core, the mode couplings can occur between modes with different azimuth variations as well as between modes with the same. Among them, the cross couplings between the fundamental core mode HEco11 and the transverse-magnetic cladding modes TMclmare most strong compared even to the self-coupling of the fundamental core mode, and thus the spectral shift with the growth of the director modulation becomes negligible. Using the derived theoretical results, we discuss numerical examples of long-period liquid-crystal fiber gratings with respect to the modulation depth of the director and the polarization states of the fundamental core mode.