A method is presented for studying harmonic wave propagation in thick circular cylinders, which are orthotropic. The analysis is carried out within the framework of the complete three-dimensional theory of elasticity. The displacements in the circumferential and longitudinal directions are taken in the form of trigonometric functions, while the radial displacement field is modelled by Frobenius power series developed through the thickness of the shell. It is shown that the Frobenius method is a powerful tool in solving for wave motions in anisotropic elastic continua. Dispersion curves are presented for axisymmetric and asymmetric waves in the case of transverse isotropy. The methodology breaks the problem down into four different tasks, which have then to be treated separately. They are as follows: the axisymmetric wave motion, n = 0; the flexural or beam-type wave motion, n = 1; the lobar wave motion, n = 2; finally the wave motion with higher circumferential wavenumbers, n q 2. Exact solutions have been found for waves travelling in thick orthotropic shells. These solutions may serve as benchmark solutions for comparison with approximate treatments of similar problems.