Relative differential cross-sections (DCS's) have been measured for the quasi-elastic scattering (elastic scattering plus rotational and vibrational excitations) of positrons from CH 4, N 2, O 2, CO, CO 2, N 2O, C 2H 2, and SF 6 molecules in a crossed-beam experiment. The energy range extends from below the positronium (Ps) formation threshold of each molecule to well above the Ps formation threshold and the angular range extends from 30° to 135°. For CH 4, N 2, CO, O 2, and SF 6 the DCS exhibits structure (a clear minimum and maximum) at low energies that tends to wash out as the energy is increased above the Ps formation threshold. In contrast, for N 2O, CO 2, and C 2H 2 we find no pronounced structure below the Ps formation threshold. These comparisons suggest that the degree of spherical symmetry of the target may be correlated with the presence of the structure.