Alternating direction iterative (ADI) methods do not usually work well on parallel computers due to having to do parallel rather than serial tridiagonal solves in all but one dimension. An ADI-like iteration is developed and analyzed which does not require parallel tridiagonal solves in any direction, has at least as good of a convergence rate as ADI, and has almost no communication when imbedded as a smoother inside of a multigrid solver. Numerical experiments on a network of workstations and a parallel computer are included.