  We characterize quasi K\"ahler manifolds whose curvature tensor associated to
the canonical Hermitian connection satisfies the first Bianchi identity. This
condition is related with the third Gray identity and in the almost K\"ahler
case implies the integrability. Our main tool is the existence of generalized
holomorphic frames introduced by the second author previously. By using such
frames we also give a simpler and shorter proof of a Theorem of Goldberg.
Furthermore we study almost Hermitian structures having the curvature tensor
associated to the canonical Hermitian connection equal to zero. We show some
explicit examples of quasi K\"ahler structures on the Iwasawa manifold having
the Hermitian curvature vanishing and the Riemann curvature tensor satisfying
the second Gray identity.
