Employing the path integral formalism, we study a quantum (test) system coupled to an environment consisting of infinitely many harmonic oscillators. A simple semiclassical approximation, in which only real classical trajectories are required, is used to derive the propagator of the reduced density matrix for Ohmic dissipation and high temperatures. The inclusion of a summation over trajectories with the correct Maslov phases in the final expression for the propagator permits investigation of nonharmonic test systems. In a numerical example for a Morse oscillator interacting with a high temperature heat bath, the semiclassical result, obtained by the method described here, correctly displays the suppression of quantum behavior as observed in the exact calculation.