  We introduce a notion of compatibility between (almost) Dirac structures and
(1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study
several properties of the "Dirac-Nijenhuis" structures thus obtained, including
their connection with holomorphic Dirac structures, the geometry of their
leaves and quotients, as well as the presence of hierarchies. We also consider
their integration to Lie groupoids, which includes the integration of
holomorphic Dirac structures as a special case.
