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

Generalized connections, spinors, and integrability of generalized structures on Courant algebroids