A loss in the effectiveness of diffusion synthetic acceleration (DSA) schemes has been observed with certain SN discretizations on two-dimensional Cartesian grids in the presence of material discontinuities. We will present more evidence supporting the conjecture that DSA effectiveness will degrade for multidimensional problems with discontinuous total cross sections, regardless of the particular physical configuration or spatial discretization. Fourier analysis and numerical experiments help us identify a set of representative problems for which established DSA schemes are ineffective, focusing on diffusive problems for which DSA is most needed. We consider a lumped, linear discontinuous spatial discretization of the SN transport equation on three-dimensional, unstructured tetrahedral meshes and look at a fully consistent and a "partially consistent" DSA method for this discretization. The effectiveness of both methods is shown to degrade significantly. A Fourier analysis of the fully consistent DSA scheme in the limit of decreasing cell optical thickness supports the view that the DSA itself is failing when material discontinuities are present in a problem. We show that a Krylov iterative method, preconditioned with DSA, is an effective remedy that can be used to efficiently compute solutions for this class of problems. We show that as a preconditioner to the Krylov method, a partially consistent DSA method is more than adequate. In fact, it is preferable to a fully consistent method because the partially consistent method is based on a continuous finite element discretization of the diffusion equation that can be solved relatively easily. The Krylov method can be implemented in terms of the original SN source iteration coding with only slight modification. Results from numerical experiments show that replacing source iteration with a preconditioned Krylov method can efficiently solve problems that are virtually intractable with accelerated source iteration.