We present DFT study on crystal and electronic structures of eleven hitherto unknown lead triiodide perovskites with unsaturated three- and four-membered heterocyclic cations containing N, O, and S atoms. According to the calculations, the proposed compounds are semiconductors with direct bandgaps varying in the range [0.84–1.61] eV. Their predicted electronic band structures feature flat bands formed from the heterocycles’ unoccupied π-MOs. With respect to the flat bands’ energies all considered perovskites can be divided into two sets. One set contains compounds whose flat bands are higher in energy than the lowest energy conduction bands formed mainly from unoccupied pi-orbitals of Pb²⁺ cations. Another set is comprised of compounds whose flat bands are the lowest energy conduction bands. Compounds described here broaden the family of hybrid metal halide perovskites. All but the lead triiodide containing unstable 2-azirinium cation are suitable for either photovoltaic or optical applications.