Fatigue life, stability and performance of majority of the structures and systems depend significantly on dynamic loadings applied on them. In many engineering cases, the dynamic loading is random vibration and the structure is a plate-like system. Examples could be printed circuit boards or jet impingement cooling systems subjected to random vibrations in harsh military environments. In this study, the response of thin rectangular plates to random boundary excitation is analytically formulated and analyzed. In the presented method, closed-form mode shapes are used and some of the assumptions in previous studies are eliminated; hence it is simpler and reduces the computational load. In addition, the effects of different boundary conditions, modal damping and excitation frequency range on dynamic random response of the system are studied. The results show that increasing both the modal damping ratio and the excitation frequency range will decrease the root mean square acceleration and the maximum deflection of the plate.