Equilibrium structures, r(e), of the heterocyclic molecules oxirane, furazan, furan, ethylene ozonide, and 1,3,4-oxadiazole have been determined using three different, somewhat complementary techniques: a completely experimental technique (r(m)), a semiexperimental technique (r(e)(SE), whereby equilibrium rotational constants are derived from experimental effective ground-state rotational constants and corrections based principally on an ab initio cubic force field), and an ab initio technique (r(e)(BO), whereby geometry optimizations are usually performed at the coupled cluster level of theory including single and double excitations augmented by a perturbational estimate of the effects of connected triple excitations [CCSD(T)] using quadruple-zeta-Gaussian basis sets). All these molecules are asymmetric tops with the moment of inertia I(c) much larger than the other two moments of inertia, I(a) and I(b). Molecules of this shape experience a large rotation of the principal axis system upon certain isotopic substitutions. For such isotopologues it is difficult to obtain a good structural fit to the semiexperimental moments of inertia I(a) and I(b) which may significantly reduce the accuracy of the r(e)(SE) structural parameters. The origin of this difficulty is explained. For the heavy-atom skeleton of these molecules it was possible to determine a rather accurate empirical mass-dependent structure without a priori knowledge of the equilibrium structure.