  We study almost K\"ahler manifolds whose curvature tensor satisfies the
second curvature condition of Gray (shortly ${\cal{AK}}_2$). This condition is
interpreted in terms of the first canonical Hermitian connection. It turns out
that this condition forces the torsion of this connection to be parallel in
directions orthogonal to the K\"ahler nullity of the almost complex structure.
We prove a local structure result for ${\cal{AK}}_2$ manifolds, showing that
the basic pieces are manifolds with parallel torsion and special almost
K\"ahler manifolds, a class generalizing, to some algebraic extent, the class
of 4-dimensional ${\cal{AK}}_2$-manifolds. In the case of parallel torsion, the
Einstein condition and the reducibility of the canonical Hermitian connection
is studied.
