The influence of heat conduction on the propagation of a surface wave polarized in the sagittal plane along the surface of a thermoelastic cylinder is investigated. The modified Maxwell law is used to take into consideration the small time that is necessary for the establishment of stationary heat conduction after the sudden occurrence of a temperature gradient in a solid. The nonstationary surface wave (Rayleigh wave) is interpreted as the line (in the given case it is a straight line parallel to the generators of the cylinder) on which temperature and the components of the stress and strain tensors experience a discontinuity. The discontinuity line propagates with a constant normal velocity in the direction of cylinder guide along the free from stresses and heat-insulated surface of the cylinder. This line is obtained by the exit onto the cylindrical surface of the three strong discontinuity complex wave surfaces intersecting along this line: quasithermal, quasilongitudinal, and quasitransverse volume waves. Applying the theory of discontinuities, the velocity and the intensity of the surface wave have been found. It has been shown that attenuation of the surface wave intensity is determined by a consideration of the related strain and temperature fields.