A Green's function approach for precisely computing the temperature and the three components of the heat flux in a rectangular parallelepiped is presented. Each face of the parallelepiped may have a different, but spatially uniform, boundary condition. Uniform volume energy generation is also treated. Three types of boundary conditions are included: type 1, a specified temperature; type 2, a specified flux; or type 3, a specified convection boundary condition. A general form of the Green's function covering all three types of boundary conditions is given. An algorithm is presented to obtain the temperature and flux at high accuracy with a minimal number of calculations for points in the interior as well as on any of the faces. Heat flux on type I boundaries, impossible to evaluate with traditional Fourier series, is found by factoring out lower-dimensional solutions. A numerical example is given. This research and resulting computer program was part of a code verification project for Sandia National Laboratories.