A free vibration analysis and an optimal design approach have been presented for thick isotropic rectangular plates with varying thickness under general edge conditions. First, the analysis is developed for vibrating rectangular plates by using the Mindlin plate theory and an eigenvalue problem is formulated by extending a method of Ritz to arbitrary sets of standard boundary conditions. The classical plate theory is also used to derive the frequency equation for comparison purpose. Secondly, a simplified optimal design approach is proposed to maximize the fundamental frequency of the plates. In applying this approach, the thickness variation is assumed to be linear in one direction and a taper ratio is chosen to be a design variable that represents the whole plate design. This approach significantly reduces the process for obtaining optimal or nearly optimal design under constraint of the constant plate volume. Numerical results are presented for various sets of boundary conditions, thickness ratio and two different plate theories, and their effects on the optimal taper ratio and its corresponding maximized fundamental frequency are discussed.