Relative differential cross sections (DCS's) have been measured for quasielastic scattering of positrons (positrons that have elastically scattered or produced rotational or vibrational excitations) from N-2, CO, O-2 N2O, and CO2, in a crossed-beam experiment. The energy range extends from below the Ps formation threshold of each molecule to 100 eV (well above the Ps formation threshold) and the angular range extends from 30 degrees to 120 degrees. For positron scattering from N-2, CO, and O-2, the DCS's exhibit structure (distinct minima and maxima) at low energies that tends to disappear as the energy is increased above the Ps formation threshold. In contrast, for N2O and CO2 we find at most only a very shallow minimum below the Ps formation threshold that also tends to disappear as the energy is increased. These observations along with prior positron-inert gas and CH4 DCS measurements suggest that the degree of spherical symmetry of the target may be correlated with the degree of prominence of the DCS structure for low energy positron scattering. Such distinctions between the diatomic and triatomic molecules have not been evident in prior corresponding electron DCS measurements. [S1050-2947(99)05407-4].