Lithium niobate (LN) exhibits outstanding properties in various application of photonics, electronics, and optoelectronics, showing potentials in integration. Due to the directional dependence of LN tensor properties, optical elements made up by LN favor the type of LN substrate. To introduce high-performance superconducting nanowire single-photon detectors to LN-integrated photonics chips, superconducting NbN thin films with thicknesses from 3 to 50 nm were deposited on X-cut, Y-cut, and Z-cut LN substrates using magnetron sputtering at room temperature. The different thickness dependencies of Tc, δTc, and residual resistance ratios are observed in NbN thin films on different LN substrates. NbN thin films on X-cut and Y-cut LN substrates are polycrystalline with a transition temperature (Tc) of ~6 K for a 6-nm-thick film. While NbN thin films are epitaxially textured on Z-cut LN substrates with Tc of 11.5 K for a 6-nm-thick film. NbN-SNSPD on X-cut LN substrates shows a weak saturation trend of its system detection efficiency; however, the performance of NbN-SNSPD on Z-cut LN substrates is limited. We evaluated the selection of cuts and concluded that X-cut and Y-cut LN are more suitable to be a platform of integrated LN photonic chips from the aspect of NbN-SNSPD. This study helps fabricate high-performance SNSPDs on fully integrated photonics chips on LN substrates.