Abstract: Based on Legendre orthogonal polynomial expansion method and the Kelvin-Voigt model, the characteristic equation for longitudinal guided waves in a viscoelastic multilayer orthotropic anisotropic hollow cylinder is derived, and an improved modulus-ratio root-seeking algorithm is proposed for solving this equation, and the physical significance of the dispersion equations is elaborated. In order to verify the reliability of the theoretical model and the developed program, the calculation of dispersion curves and attenuation curves for a large outer radius-to-thickness ratio viscoelastic single-layer tube is firstly carried out and compared with the published numerical results. Subsequently, the effects of the outer radius-to-thickness ratio and the viscosity constant on the dispersion and attenuation curves are investigated using a three-layer viscoelastic hollow cylinder as an example. Finally, the calculation of three-dimensional complex wave number-frequency dispersion curves is carried out and the non-propagating modes are analyzed.