Exact expressions are given for the free energy of the classical Heisenberg model in one lattice dimension and two or three spin dimensions with general anisotropy; the two-dimensional result, already known, is in terms of Mathieu functions and the three-dimensional result in terms of Lame wavefunctions. The method presented solves the eigenvalue problem for an appropriate transfer matrix by relating it to solutions of the Helmholtz equation in suitable curvilinear coordinate systems. High and low temperature limits are discussed for the three-dimensional spin case and the connection given with already known results for partial anisotropy.