There have been significant research works on the free vibration behavior of plates with holes owing to its importance in engineering applications such as aerospace, civil, marine, and mechanical engineering fields. It is well known that the inclusion of holes in plate structure changes the dynamic behavior of such plates and gives rise to many difficulties in numerical studies especially for three-dimensional cases. In the current paper, the free vibration behavior of homogeneous rectangular plates with a centrally placed circular hole is analyzed based on the three-dimensional elasticity solution along with the meshless technique known as the Radial Point Interpolation Method (RPIM). The approximation scheme of displacement field in terms of RPIM shape functions is made based on multiquadric radial basis functions combined with polynomial basis functions. The derivation of the frequency equation for motion is performed through the principle of stationary energy and the linear elastic governing differential equations. Contrary to several other conventional techniques developed for perforated plates, the present approach does not require any domain decomposition in the neighborhood of the hole or subtracting the hole energy from total strain energy. A MATLAB code is written for the implementation of the proposed formulation. The computed values are verified using finite element method performed by ABAQUS as well as other benchmark solutions presented in literature. A detailed study on the effect of cutout size and boundary conditions on the natural frequency of the plates is carried out. The results clearly indicate the accuracy, robustness, and effectiveness of the proposed meshless three-dimensional formulation.