We present a discontinuous Galerkin method for the plate problem. The method employs a discontinuous approximation space allowing nonmatching grids and different types of approximation spaces. Continuity is enforced weakly through the variational form. Discrete approximations of the normal and twisting moments and the transversal force, which satisfy the equilibrium condition on an element level, occur naturally in the method. We show optimal a priori error estimates in various norms and investigate locking phenomena when certain stabilization parameters tend to infinity. Finally, we relate the method to two classical elements: the nonconforming Morley element and the C-1 Argyris element.