We study theoretically persistent currents in rings made of band insulators with a fully occupied valence band and empty conduction band. We derive an expression for the persistent current in a one-dimensional (1D) insulating ring with an arbitrary valence band ε(k). Setting for ε(k) the realistic (numerically calculated) valence bands, we estimate persistent currents in the nanorings made of real band insulators like GaAs, Ge, and InAs. Further, we express the persistent current in the insulating 1D ring through the Wannier functions of the constituting infinite 1D crystal. We find these Wannier functions analytically in the LCAO approximation. We obtain fully analytically the persistent current in the insulating ring and we apply the result to a 1D GaAs ring with conduction electrons subjected to a periodic potential emulating the insulating lattice. In general, persistent currents in rings made of band insulators decay with the ring length exponentially due to the exponential decay of the Wannier function tails. In spite of that, the currents of measurable size are found for a suitably chosen nanorings similar to those fabricated recently for coherent nanoelectronics. Finally, to provide insight, our major results are rederived from the so-called ring Wannier functions.