This paper presents a free vibration analysis of thick arbitrary quadrilateral plates based on the Mindlin shear deformation theory. The recently developed pb-2 Ritz method has been employed to compute the solution. The method features the versatile pb-2 Ritz functions defined by the product of a two-dimensional polynomial (p-2) and a basic function (b) to approximate the deflections and rotations. The arbitrary thick quadrilateral plate is first mapped onto a basic square plate. The basic function is then formed from the product of boundary expressions of the basic square plate, each raised to an appropriate power to satisfy the various geometric boundary conditions of the actual plate. Stiffness and mass matrices are numerically integrated over the domain of the basic plate by using Guassian quadrature. New sets of natural frequency parameters for quadrilateral Mindlin plates with different combinations of boundary conditions and various thickness-to-width ratios are presented. Wherever possible, the present results are verified by comparison with existing analytical and experimental values from open literature. The results from the pb-2 Ritz method are also directly compared with those determined by using the finite element method. The effect of the Poisson ratio on the natural frequencies of a thick cantilever quadrilateral plate is also investigated.