A multigrid acceleration scheme for the one-dimensional slab geometry SN equations with anisotropic scattering and linear discontinuous spatial differencing is developed. The high-frequency relaxation iteration consists of three steps: a standard source iteration, independent two-cell block inversions centered about each spatial cell edge, and an averaging of the iterates from the previous two steps. Because the linear discontinuous differencing scheme is a finite element method, fine-to-coarse projection and coarse-to-fine interpolation are straightforward. Although standard linear discontinuous differencing is derived under the assumption of spatially constant cross sections within each cell, the scheme is generalized to allow for a linear spatial variation of the cross section in each cell. This linear variation is required to obtain accurate coarse-grid equations when homogenizing two fine-grid cells with different cross sections into a single coarse-grid cell. This multigrid method is very effective in terms of the spectral radius of the total iteration process, but the computational cost of the block inversions in the second step of the high-frequency relaxation is quite high. However, in optically thick problems with highly anisotropic scattering, this multigrid method is more economical than diffusion synthetic acceleration. Because the block inversions are independent for each cell edge, parallel processing might significantly reduce the cost of the scheme.