Recent work has identified a non-zinc-blende-type quaternary semiconductor, Cu2BaSnS4–x Se x (CBTSSe), as a promising candidate for thin-film photovoltaics (PVs). CBTSSe circumvents difficulties of competing PV materials regarding (i) toxicity (e.g., CdTe), (ii) scarcity of constituent elements (e.g., Cu­(In,Ga)­(S,Se)2/CdTe), and (iii) unavoidable antisite disordering that limits further efficiency improvement (e.g., in Cu2ZnSnS4–x Se x ). In this work, we build on the CBTSSe paradigm by computationally scanning for further improved, earth-abundant and environmentally friendly thin-film PV materials among the 16 quaternary systems I2–II–IV–VI4 (I = Cu, Ag; II = Sr, Ba; IV = Ge, Sn; VI = S, Se). The band structures, band gaps, and optical absorption properties are predicted by hybrid density-functional theory calculations. We find that the Ag-containing compounds (which belong to space groups I222 or I4̅2m) show indirect band gaps. In contrast, the Cu-containing compounds (which belong to space group P31/P32 and Ama2) show direct or nearly direct band gaps. In addition to the previously considered Cu2BaSnS4–x Se x system, two compounds not yet considered for PV applications, Cu2BaGeSe4 (P31) and Cu2SrSnSe4 (Ama2), show predicted quasi-direct/direct band gaps of 1.60 and 1.46 eV, respectively, and are therefore most promising with respect to thin-film PV application (both single- and multijunction). A Cu2BaGeSe4 sample, prepared by solid-state reaction, exhibits the expected P31 structure type. Diffuse reflectance and photoluminescence spectrometry measurements yield an experimental band gap of 1.91(5) eV for Cu2BaGeSe4, a value slightly smaller than that for Cu2BaSnS4.