This paper is concerned with a simplified dynamical analysis of orthotropic viscoelastic plates that are made up of an arbitrary number of layers each of which is a Maxwell type solid. This study includes the case where some or all the layers are themselves constituted by thinly laminated materials with couple stresses. The recurrence equations for the shear stresses are obtained for an arbitrary number of layers and then applied to plates with two or three layers. The viscoelastic damping effect is determined by the process of linearization and then illustrated by a plate composed of one, two or three layers. It is found that the damping increases with anisotropy and wave number. These results are shown by graphical representations.