This study investigates the attenuation of the seismic Rayleigh waves propagating in an elastic crustal layer of the Earth over its viscoelastic mantle. The exact equations of motion of the theory of linear viscoelasticity are used and the complex dispersion equation is obtained for an arbitrary type of hereditary operator of the viscoelastic materials. The viscoelasticity of the materials is described by the fractional-exponential operators of Rabotnov, and a solution algorithm is developed to obtain numerical results for the attenuation of the considered waves. Attenuation curves are obtained and discussed, and in particular, the influence of the rheological parameters of the materials on this attenuation is studied. It is established that a decrease in the creep time of the viscoelastic materials leads to an increase in the attenuation coefficient.