In this article we present a high-order-accurate solver for the radiative transfer equation (RTE) which uses the discontinuous Galerkin (DG) method and is designed for graphics processing units (GPUs). The compact nature of the high-order DG method enhances scalability, particularly on GPUs. High-order spatial accuracy can be used to reduce discretization errors on a given computational mesh, and can also reduce the mesh size needed to achieve a desired error tolerance. Computational efficiency is a key concern in solutions to radiative heat transfer problems, due to potentially large problem sizes created by (a) the presence of participating nongray media in a full-spectrum analysis, (b) the need to resolve a large number of angular directions and spatial extent of the domain for an accurate solution, and (c) potentially large variations in material and flow properties in the domain. We present here a simulation strategy, as well as a set of physical models, accompanied by a number of case studies, demonstrating the accuracy and superior performance in terms of computational efficiency of this approach.