An analytical-numerical method is presented for analysing characteristics of waves in a cylinder composed of functionally graded material (FGM). In this method, the FGM cylinder is divided into a number of annular elements with three-nodal-lines in the wall thickness. The elemental material properties are assumed to vary linearly in the thickness direction so as to better model the spatial variation of material properties of FGM. The Hamilton principle is used to develop the dispersion equations for the cylinder, and the frequency and the group velocity are established in terms of the Rayleigh quotient. The method is applied to analyse several FGM cylinders, and its effciency is demonstrated. Numerical results demonstrate that the ratio of radius to thickness has a stronger influence on the frequency spectra in the circumferential wave than on that in the axial wave, that negative group velocity presents at a range of smaller wave numbers and that the range varies as the wave normal and the ratio of radius to thickness of FGM cylinders.