Given a complex polynomial P with zeroes z(1),..., z(d), we show that the asymptotic zero-counting measure of the iterated derivatives Q((n)), n = 1, 2,..., where Q = R/P is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with z(1),...,z(d). This refines Polya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in C-m.