Presented herein is a novel computational approach using the moving element method (MEM) for simulating the dynamic response of Mindlin plate resting on a viscoelastic foundation and subjected to moving loads. The governing equations and the element mass, damping and stiffness matrices are formulated in a convected coordinate system in which the origin is attached to the point of the moving applied load. Thus, the method simply treats moving loads as stationary at the nodes of the plate to avoid updating the locations of moving loads due to the change of the contact points on the plate. To verify the accuracy of the proposed computational approach, static and free vibration analyses of plates are investigated first. Next, the dynamic response of plate resting on a viscoelastic foundation subjected to a moving load is examined. A parametric study is performed to determine the effects of the loads velocity, foundation damping and foundation stiffness on the dynamic response of a plate. Finally, the comparisons of the dynamic response of plates resting on viscoelastic foundation and subjected to moving vehicles with three models of load (single-wheel, single-axle and tandem-axle) are discussed.