A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed method equations for the biharmonic Dirichlet problem is presented. In particular, a Schur complement formulation for these equations which yields non-inherited quadratic forms is considered. The preconditioning scheme is compared with a standard W-cycle multigrid iteration. It is proved that a Variable V-cycle preconditioner leads to problems with uniformly bounded condition numbers. However, W-cycle convergence is proved only if the number of smoothings "m is sufficiently large". An example is given in which the W-cycle diverges unless "m > 7". Divergent W-cycles are also encountered when solving the Morley equations for the biharmonic Dirichlet problem; although, Brenner has proved W-cycle convergence for sufficiently large "m" [Brenner,(1989)]. This is illustrated with additional computations, while Variable V-cycles continue to produce excellent preconditioners in this setting. Certain approximate L2-inner products are described and a modification to the Ciarlet-Raviart method is proposed which reduces the work of the multilevel schemes. Optimal order error estimates are proved for the modified method. Consideration is restricted to Ciarlet-Raviart methods of quadratic and higher degree throughout the paper.