We study the conformational behavior of DNA molecules adsorbed on cationic-lipid membranes that are supported on grooved, one-dimensionally periodic microstructured surfaces. We reveal a striking ability of these periodically structured membranes to stretch DNA coils. We elucidate this DNA stretching phenomenon in terms of surface curvature dependent potential energy attained by the adsorbed DNA molecules. Because of it, DNA molecules undergo a localization transition causing them to stretch by binding to highly curved sections of the supported membranes.