<p>Multilayered plate and shell structures are widely used in mechanical engineering and aerospace engineering. The paper investigates the propagation of non-axisymmetric waves in a three-layered viscoelastic cylindrical shell. The displacements of the outer shell are described using the shell equations based on the Kirchhoff–Love hypothesis, while the displacements of the middle layer (or filler) are modeled using the equations of viscoelasticity theory in a polar coordinate system. The primary objective of this study is to derive the dispersion equation and conduct a numerical analysis of the damping coefficient as a function of the mechanical system’s parameters. Methods from elasticity theory and computational mathematics, including techniques by Müller, Gauss, and Laplace, are employed to solve the problem. In the non-axisymmetric case, longitudinal–transverse and torsional waves are coupled, with their propagation velocities determined from a unified dispersion equation. The findings reveal that as the filler thickness increases, the real and imaginary parts of the phase velocities for the first mode increase, while those for the second mode decrease.</p>