On-chip photonic quantum circuits with integrated quantum memories have the potential to radically advance hardware for quantum information processing. In particular, negatively charged group-IV color centers in diamond are promising candidates for quantum memories as they combine long storage times with excellent optical emission properties and an optically addressable spin state. However, as a material, diamond lacks the many functionalities needed to realize scalable quantum systems. Thin-film lithium niobate (TFLN), in contrast, offers a number of useful photonic nonlinearities, including the electro-optic effect, piezoelectricity, and capabilities for periodically poled quasi-phase matching. Here, we present the highly efficient heterogeneous integration of diamond nanobeams containing negatively charged silicon-vacancy (SiV) centers with TFLN waveguides. We observe greater than 90% transmission efficiency between the diamond nanobeam and the TFLN waveguide on average across multiple measurements. By comparing saturation signal levels between confocal and integrated collection, we estimate a more than 10-fold increase in photon emission channeled into TFLN waveguides versus that channeled into out-of-plane collection channels. Our results constitute a key step for creating scalable integrated quantum photonic circuits that leverage the advantages of both diamond and TFLN materials.