In this paper, three dimensional (3D) static and dynamic analysis of thick plates based on the Meshless Local Petrov-Galerkin (MLPG) is presented. Using the kinematics of a three-dimensional continuum, the local weak form of the equilibrium equations is derived. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains by using a unit test function. Nodal points are distributed in the 3D analysis domain and each node is surrounded by a cubic sub-domain to which a local integral equation is applied. The meshless approximation based on the three dimensional Moving Least-Square (MLS) is employed as the shape function to approximate the field variable of scattered nodes in the problem domain. The Newmark time integration method is used to solve the system of coupled second order ODEs. The essential boundary conditions are enforced by the direct interpolation method. Numerical examples for solving the static and transient response of elastic thick plates are demonstrated. The numerical efficiency of the proposed meshless method is demonstrated by comparing the results obtained with the available analytical and/or numerical solutions in the literature.