Elastodynamic response of an undamped moderately thick plate, with arbitrary boundary conditions, under a moving mass is investigated. The FSDT (first-order shear deformation plate theory or Mindlin plate theory) is selected as the governing equations of motion. By using direct separation of variables and eigenfunction expansion method, the three basic variables defining the displacement field in FSDT, are transformed into a series including the eigenfunctions of plate free vibration with time dependent amplitude factors. By neglecting the inertia interaction between mass and the plate, the closed-form solution is derived while it remarkably reduces the complication of numerical computations. Having the moving mass inertia effect taken into account as well as all the convective terms of its out-of-plane acceleration components, a semi analytical solution is presented. The most interested moving mass trajectories in engineering application of the issue, orbiting and rectilinear paths are investigated in numerical examples and the results for a simply supported rectangular Mindlin plate are obtained. The method introduced is not limited by shape of the plate and trajectory of the moving mass. Concentrated moving loads as well as other arbitrarily selected distribution-area of loads are covered in the formulations. Parametric survey is carried out by using both FSDT and CPT (classical plate theory or Kirchhoff plate theory) and remarkable differences between CPT and FSDT modeling results, for moderately thick plates, emphasize the significance of using FSDT. For thin plates, the FSDT yields closely near the same results as that of CPT which demonstrates the generality of the solutions presented in this article with regard to capability of capturing the under study plate dynamics for a wider range of the plate thickness with appropriate precision.