Common materials, ranging from natural wood to modern composites, have been recognized as ortho-tropic materials. The elastic properties of such materials are governed by nine elastic constants. In this paper the complete set of Green's functions for an infinite medium and a half space is given, which were not reported completely before. Analytic expressions for the infinite Green's functions are derived through the explicit form of the sextic equation given explicitly. For an orthotopic half space, the Green's function is derived by a superposition method. The mathematical concept is based on the addition of a complementary term to the Green's function in an orthotropic infinite domain to fulfill the boundary condition on the free surface. Both solutions are illustrated in certain directions to demonstrate the nature of orthotropy.