A semiclassical approximation is proposed for the Monte Carlo quantum mechanical path integral approach to the calculation of equilibrium properties (partition functions, free energies) of molecules or generalized transition states. The approximation is based on replacing the sum over paths by a sum over points that are displaced from the classically sampled points. The displacements are adjusted to ensure the correct results for harmonic potentials, and thus the method is akin to perturbation theory with a harmonic reference. In its simplest form, the approximation takes the form of a classical mechanical calculation with double sampling for every configuration point; thus it can be readily adapted to a variety of applications. Calculations by several other approximate methods are also presented, and the results with the new method compare very favorably with them.