Searches for low-energy tetrahedral polymorphs of carbon and silicon have been performed using density functional theory computations and the ab initio random structure searching approach. Several of the hypothetical phases obtained in our searches have enthalpies that are lower or comparable to those of other polymorphs of group 14 elements that have either been experimentally synthesized or recently proposed as the structure of unknown phases obtained in experiments, and should thus be considered as particularly interesting candidates. A structure of Pbam symmetry with 24 atoms in the unit cell was found to be a low-energy, low-density metastable polymorph in carbon, silicon, and germanium. In silicon, Pbam is found to have a direct band gap at the zone center with an estimated value of 1.4 eV, which suggests applications as a photovoltaic material. We have also found a low-energy chiral framework structure of P41212 symmetry with 20 atoms per cell containing fivefold spirals of atoms, whose projected topology is that of the so-called Cairo-type two-dimensional pentagonal tiling. We suggest that P41212 is a likely candidate for the structure of the unknown phase XIII of silicon. We discuss Pbam and P41212 in detail, contrasting their energetics and structures with those of other group 14 elements, particularly the recently proposed P42/ncm structure, for which we also provide a detailed interpretation as a network of tilted diamondlike tetrahedra.