Clusters of workstations have become a cost-effective means of performing scientific computations. However, large network latencies, resource sharing, and heterogeneity found in networks of clusters and Grids can impede the performance of applications not specifically tailored for use in such environments. A typical example is the traditional fine grain implementations of Krylov-like iterative methods, a central component in many scientific applications. To exploit the potential of these environments, advances in networking technology must be complemented by advances in parallel algorithmic design. In this paper, we present an algorithmic technique that increases the granularity of parallel, block iterative methods by inducing additional work during the preconditioning (inexact solution) phase of the iteration. During this phase, each vector in the block is preconditioned by a different subgroup of processors, yielding a much coarser granularity. The rest of the method comprises a small portion of the total time and is still implemented in fine grain. We call this combination of fine and coarse grain parallelism multigrain. We apply this idea to the block Jacobi-Davidson eigensolver, and present experimental data that shows the significant reduction of latency effects on networks of clusters of roughly equal capacity and size. We conclude with a discussion on how multigrain can be applied dynamically based on runtime network performance monitoring.