The discrete transfer method is extended to solve the radiative transfer equation in two- and three-dimensional semi-transparent media with variable refractive index. The radiative transfer equation and other related equations are organized based on the refracted quantities. A cell-based ray tracing approach is used to track the curve paths of radiative rays in the medium. The complex geometries in multi-dimensional media are simulated by a simple blocked-off strategy. The present method is verified by comparing its results with benchmark solutions, and the performance of the method is examined by some numerical experiments. (C) 2017 Elsevier Ltd. All rights reserved.