A multigrid method is applied to a fourth-order accurate (spatially and temporally) finite difference Runge-Kutta time-marching scheme. This is used to solve the nonlinear Euler equations for the efficient prediction of noise radiation from turbofan engine inlets. This noise prediction approach computes a steady-state solution first, and then a source is turned on and the unsteady solution is computed. This has the advantage of using the same numerical scheme to evaluate the residuals of the governing equations in both the steady and unsteady calculations, which means that the steady-state solution is smooth and no numerical errors contaminate the acoustic solution. The high resolution steady-state flowfields are calculated efficiently using a full approximation storage multigrid method. This makes it to possible to attain steady-state solutions on extremely fine meshes designed for high-frequency turbofan noise problems. It is also shown that acoustic results for a JT15D inlet are accurately represented by the present approach.