We study the dynamics of a layer of viscoelastic fluid, in the Stokesian regime, that is driven from below by a 4 x 4 checkerboard pattern of rotating and counter-rotating disks. At low disk rotation rate (low Weissenberg number) the fluid flow response is slaved to the geometry of this forcing and divides into many steadily rotating cells, each contained within invariant manifolds issuing from hyperbolic stagnation points. As the rotation rate increases these fluid cells begin to oscillate periodically in a synchronized fashion. At a yet higher rotation rate, this temporally periodic flow disappears and is replaced by a richer, "turbulent" dynamics where the flow is delocalized from the forcing and has fluid cells that are continuously destroyed and reformed. Published by Elsevier B.V.