A dual time algorithm for the compressible Navier-Stokes equations requires an efficient and reliable subiterative process to make it effective for solving a wide range of timedependent flows. For the subiterative process we consider a multigrid method with a Runge-Kutta (RK) scheme that is accelerated by a fully implicit preconditioner. This particular approach has been shown to be highly effective for steady-state problems. It allows high CFL numbers while retaining good high frequency smoothing properties for the RK scheme. We derive in detail the dual time algorithm and use Fourier analysis to show the properties of the subiterative method. To demonstrate the efficiency of the dual time algorithm we consider unsteady flows about an airfoil. These results clearly demonstrate the vast improvement over existing dual time algorithms relying upon RK schemes. The present algorithm has the distinct advantage that it can be readily incorporated into the large number of existing computer codes that use RK schemes.