Algorithm and code are presented that solve dispersion equations for cylindrically layered media consisting of an arbitrary number of elastic and fluid layers. The algorithm is based on the spectral method which discretizes the underlying wave equations with the help of spectral differentiation matrices and solves the corresponding equations as a generalized eigenvalue problem. For a given frequency the eigenvalues correspond to the wave numbers of different modes. The advantage of this technique is that it is easy to implement, especially for cases where traditional root-finding methods are strongly limited or hard to realize, i.e., for attenuative, anisotropic, and poroelastic media. The application of the new approach is illustrated using models of an elastic cylinder and a fluid-filled tube. The dispersion curves so produced are in good agreement with analytical results, which confirms the accuracy of the method. Particle displacement profiles of the fundamental mode in a free solid cylinder are computed for a range of frequencies.