Low-energy structures of alanates are currently known to be described by patterns of isolated, nearly ideal tetrahedral [AlH4] anions and metal cations. We discover that the novel polymeric motif recently proposed for LiAlH4 plays a dominant role in a series of alanates, including LiAlH4, NaAlH4, KAlH4, Mg(AlH4)2, Ca(AlH4)2, and Sr(AlH4)2. In particular, most of the low-energy structures discovered for the whole series are characterized by networks of corner-sharing [AlH6] octahedra, forming wires and/or planes throughout the materials. Finally, for Mg(AlH4)2 and Sr(AlH4)2, we identify two polymeric phases to be lowest in energy at low temperatures.