The dynamic response of cylindrical panels can be obtained analytically in very few (and simple) cases. For complicated cylinders, researchers resort to the finite element (FE) method. This can lead to large models, especially at higher frequencies, which translates into high computation costs. In this paper, the response of such structures is obtained using a wave and finite element (WFE) technique. Noting the homogeneity of cylinders around the circumference and along the axes, the FE model of a small segment of the cylinder can be post-processed using periodic structure theory to yield the wave characteristics of the cylinder. Thus, cylinders with arbitrary complexity can be considered since the full power of FE methods can be utilised in obtaining the FE model of the small segment. Then, the response of cylinder is posed as an inverse Fourier transform. Since the wavelength (i.e., the wavenumber) around the circumference can be an integer number only, one of the integrals in the inverse Fourier transform becomes a simple summation, whereas the other can be resolved analytically using contour integration and the residue theorem. The result is a computationally efficient method for obtaining the response of the cylinder to an arbitrarily distributed load.