The paper studies the dispersion of axisymmettic longitudinal waves in the bi-material compound circular cylinder made of linear viscoelastic materials. The investigations are carried out within the scope of the piecewise homogeneous body model by utilizing the exact equations of linear viscoelasto-dynamics. The corresponding dispersion equation is derived for an arbitrary type of hereditary operator and the algorithm is developed for its numerical solution. Concrete numerical results are obtained for the case where the relations of the constituents of the cylinder are described through fractional exponential operators. The influence of the viscosity of the materials of the compound cylinder on the wave dispersion is studied through the theological parameters which indicate the characteristic creep time and long-term values of the elastic constants of these materials. Dispersion curves are presented for certain selected dispersive and non-dispersive attenuation cases under various values of the problem parameters and the influence of the aforementioned theological parameters on these curves is discussed. As a result of the numerical investigations, in particular, it is established that in the case where the rheological parameters of the components of the compound cylinder are the same, the viscosity of the layers' materials causes the axisymmetric wave propagation velocity to decrease.