The contact problem of constant velocity sliding of the doubly periodic wavy surface of a rigid indentor along a viscoelastic layer on a rigid foundation is considered with the adhesive attraction between the surfaces taken into account. A method is proposed for calculating the configuration of the contact regions and adhesive interaction regions, the pressure distribution on the contact surface, and the deformation component of the friction force for various values of the viscosity, adhesion, velocity, and microgeometry parameters of the surface. We study how these parameters affect the merging of contact spots and the transition from discrete to saturated contact as the external load increases.