The results of high-throughput screening of the inorganic crystal structure database for new promising Ca2+-, Mg2+-, Zn2+- and Al3+-ion conducting ternary and quaternary sulfides, selenides, and tellurides are presented (∼1500 compounds). A geometrical-topological approach based on the Voronoi partition was initially used and yielded 104 compounds, which were unknown as conductors with possible cation migration. All compounds were passed through the bond valence site energy analysis to determine the migration energy Em. Furthermore, we established the logarithmic dependencies of Em on the geometrical parameters of the migration pathways. As a result, 16 out of 104 structures were filtered out as promising conductors. Finally, density functional theory simulations yielded the 11 most prospective compounds with Em < 1.0 eV. Among them, we found a novel class of ionic conductors with the La3CuSiS7 structure, for which ab initio molecular dynamic calculations were performed, revealing diffusion coefficients of ∼10-7 cm2 s-1 and ionic conductivity of ∼10-2 S cm-1 at 300 K.