Maxwell's equations for a laterally periodic inhomogeneous structure are considered. The transmission Green's function approach based on a vacuum wave-splitting is used to solve the scattering problem. A dielectric grating is considered first. The direction of the incident plane wave is arbitrary, and numerical results for the co- and cross-polarized reflection and transmission for TE and TM waves are presented. The approach is also generalized to the scattering from a bi-anisotropic grating.