The purpose of this paper is to present a new technique for accelerating thermal upscatter source iterations in multigroup neutron transport calculations. This technique is based upon the use of a one-group diffusion equation to approximate the multigroup $S_n$ transport equation. It can be thought of as a two-grid technique since the diffusion operator represents a coarse-grid approximation to the transport operator. In particular, the diffusion operator is coarse with respect to the direction and energy variables, but it is full-rank with respect to the spatial variables. Thermal upscatter iterations are often very slow to converge in problems containing materials such as heavy-water and graphite. For instance, using infinite-medium Fourier analyses in conjunction with a 69-group neutron cross-section set having 41 thermal energy groups, we have calculated unaccelerated spectral radii of 0.9998 and 0.998 for heavy-water and graphite, respectively. The corresponding accelerated spectral radii are 0.46 and 0.62, respectively. Thus the spectral radius is dramatically reduced with a single coarse-grid equation. Computational results are presented which indicate that our method is both efficient and robust.