A new complex-variable formalism for the analysis of three-dimensional (3D) steady-state heat transfer problems in homogeneous solids with general anisotropic behaviour is proposed in this paper. The derived method is based on the Radon transform, which is used in order to reduce the dimension of the problem to a two-dimensional (2D) Radon space where a solution can be easily handled via a complex-variable method. Subsequently, the 3D solution is obtained by applying the inverse Radon transform. Despite that the main goal of this work is to develop and illustrate the general methodology, the proposed formalism is further applied to derive new Green's functions as application examples. Contributions include new forms for bimaterial and half-space Green's functions for line, point, vortex and dipole heat sources. In particular Green's functions due to heat vortex loop sources for infinite media, half-space and bimaterial systems are presented for the first time. The veracity and computability of the approach are demonstrated with some numerical examples. (C) 2014 Elsevier Ltd. All rights reserved.