Magnetic domain wall motion could be the key to the next generation of data storage devices, shift registers without mechanically moving parts. Various concepts of such so-called ‘racetrack memories’ have been developed, but they are usually plagued by the need for high current densities or complex geometrical requirements. We introduce a new device concept, based on the interfacial Dzyaloshinskii-Moriya interaction (DMI), of which the importance in magnetic thin films was recently discovered. In this device the domain walls are moved solely by magnetic fields. Unidirectionality is created utilizing the recent observation that the strength with which a domain wall is pinned at an anisotropy barrier depends on the direction of the in-plane field due to the chiral nature of DMI. We demonstrate proof-of-principle experiments to verify that unidirectional domain-wall motion is achieved and investigate several material stacks for this novel device including a detailed analysis of device performance for consecutive pinning and depinning processes.