Small isolated clusters of alkali halides and analogous divalent species composed of closed-shell atomic ions are known to prefer densely packed structures similar but not identical to the repent units found in the solid state. Beyond the simple cube structure that occurs, for example, in alkali halide tetramers, no other ionic polyhedral structures containing only atomic ions are known. In this article, we explore the possibility of other three-dimensional structures, and we show computationally that two molecules, Mg2Al4O8 and Na4Mg4O6, have very stable distorted rhombic dodecahedron structures. We therefore suggest that a wide variety of stable polyhedral structures could be constructed from ions, and we offer a simple electrostatic model for predicting which polyhedral ionic structures should be stable and which should not. We Feint out that several recently observed inorganic solid state materials contain core structures that fit our predictions.