Reported are 11 new ternary phases with the general formula RE(7)Zn(21)Tt(2) (RE = La-Nd; Tt = Ge, Sn, and Pb), synthesized from the respective elements by reactions at high temperature. Their structures, established on the basis of single-crystal and powder X-ray diffraction work, are shown to be a new structure type with the orthorhombic space group Pbam (No. 55, Pearson symbol oP60). This complex atomic arrangement features condensed polyhedra made up of Zn atoms, interspersed by Ge, Sn, or Pb atoms in trigonal-planar coordination. The structure bears resemblance with the La3Al11 and the LaRhSn2 structure types, which are compared and discussed. Temperature dependent dc magnetization measurements confirm RE3+ ground states for all rare-earth elements, and the expected local-moment magnetism due to the partial filling of their 4f states for RE3+ = Ce3+, Pr3+, and Nd3+. Theoretical considerations of the electronic structure based on the tight-binding linear muffin-tin orbital (TB-LMTO-ASA) method are also presented: the calculations support the experimental observation of a small, but not negligible, homogeneity range in RE(7)Zn(21+x)Tt(2-x), (x < 0.5). The partial substitution of the tetrel atoms by the electron-poorer Zn appears to be an important attribute, leading to an optimal valence electron concentration and, thereby, to the overall electronic stability of the crystal structure of this family of polar intermetallics.