The problems of linear thermoelasticity in a transversely isotropic hollow cylinder of finite length are solved by a direct power series approximation through the application of the Lanczos-Chebyshev method. Coefficients in the “economized” series are determined by collocation at selected Chebyshev points. Formulations are given to show that the same approach can be applied to all linear types of boundary conditions. A numerical example has been used for comparison with pubished solutions from a potential function approach, and with “long cylinder” solutions. The magnitude of end effects has been shown.