This paper deals with a three‐dimensional elasticity investigation of wave propagation in an infinitely long hollow cylinder whose homogeneous or layered structure is made by generally orthotropic or monoclinic materials. The analysis is based on a recursive formulation of a successive approximation method employed elsewhere by the authors for the study of axisymmetric and flexural vibrations of homogeneous orthotropic and cross‐ply laminated hollow cylinders of finite length. The method has further been applied in connection with static and thermoelastic analyses of orthotropic homogeneous and cross‐ply laminated cylinders and cylindrical panels. In all cases considered, results of this approach converge very rapidly toward corresponding exact three‐dimensional elasticity results. Hence the present study is based on essentially exact three‐dimensional elasticity results and is further expected to serve as a benchmark for corresponding future investigations based on finite element and/or two‐dimensional shell theory formulations.