  We present a characterization, in terms of torsion-free generalized
connections, for the integrability of various generalized structures
(generalized almost complex structures, generalized almost hypercomplex
structures, generalized almost Hermitian structures and generalized almost
hyper-Hermitian structures) defined on Courant algebroids. We develop a new,
self-contained, approach for the theory of Dirac generating operators on
regular Courant algebroids with scalar product of neutral signature. As an
application we provide a criterion for the integrability of generalized almost
Hermitian structures (G, \mathcal J) and generalized almost hyper-Hermitian
structures (G, \mathcal J_{1}, \mathcal J_{2}, \mathcal J_{3}) defined on a
regular Courant algebroid E with scalar product of neutral signature, in terms
of canonically defined differential operators on spinor bundles associated to
E_{\pm} (the subbundles of E determined by the generalized metric G).
