The Alternating Anderson-Richardson (AAR) method provides an efficient and scalable alternative to current state-of-the-art preconditioned Krylov solvers for the solution of large, sparse linear systems on high performance computing platforms, with increasing advantage as the number of processors is increased. Moreover, the method is simple and general, applying to symmetric and nonsymmetric systems, real and complex alike. Preconditioners that are used for other linear solvers can also be used with the AAR method.