In this paper, the dynamic response of a double-beam system traversed by a constant moving load is studied. The system consists of two elastic homogeneous isotropic Euler–Bernoulli beams. The two simply supported prismatic beams are identical, parallel one upon the other and connected continuously by a viscoelastic layer. The dynamic deflections of both beams are given in analytical closed forms. The effects of the moving speed of the load and the damping and the elasticity of the viscoelastic layer on the dynamic responses of the beams are investigated in detail. Also several plots of the deflections of the beams are given and discussed for different values of speed parameter, damping ratio, and stiffness parameter. Furthermore, the force transmitted between the beams is determined and the effects of the system parameters on this force are studied. Plots of the maximum responses of the beams are presented.