  A method for calculating the retarded Green's function for the gravitational
wave equation in Friedmann-Roberson-Walker spacetimes, within the formalism of
linearized Einstein gravity is developed. Hadamard's general solution to
Cauchy's problem for second-order, linear partial differential equations is
applied to the FRW gravitational wave equation. The retarded Green's function
may be calculated for any FRW spacetime, with curved or flat spatial sections,
for which the functional form of the Ricci scalar curvature $R$ is known. The
retarded Green's function for gravitational waves propagating through a
cosmological fluid composed of both radiation and dust is calculated
analytically for the first time. It is also shown that for all FRW spacetimes
in which the Ricci scalar curvatures does not vanish, $R \neq 0$, the Green's
function violates Huygens' principle; the Green's function has support inside
the light-cone due to the scatter of gravitational waves off the background
curvature.
