We examine the dynamics of DNA molecules in mixed flows where the ratio of vorticity to strain rate may be slightly above or below unity via Brownian dynamics simulation. We find that the chain dynamics in these flows are dramatically different than those found for simple shear flow. When the strain rate exceeds vorticity, the dynamics are found to be driven by the extra amount of straining. For vorticity-dominated flows, a periodicity in chain extension is observed with considerable chain deformation.