Relativistic Green functions in a plane wave gravitational background

We consider a massive relativistic particle in the background of a gravitational plane wave. The corresponding Green functions for both spinless and spin 1/2 cases, previously computed by A. Barducci and R. Giachetti \cite{Barducci3}, are reobtained here by alternative methods, as for example, the Fock-Schwinger proper-time method and the algebraic method. In analogy to the electromagnetic case, we show that for a gravitational plane wave background a semiclassical approach is also sufficient to provide the exact result, though the lagrangian involved is far from being a quadratic one.