  The super or Z_2-graded Schouten-Nijenhuis bracket is introduced. Using it,
new generalized super-Poisson structures are found which are given in terms of
certain graded-skew-symmetric contravariant tensors \Lambda of even order. The
corresponding super `Jacobi identities' are expressed by stating that these
tensors have zero super Schouten-Nijenhuis bracket with themselves
[\Lambda,\Lambda]=0. As a particular case, we provide the linear generalized
super-Poisson structures which can be constructed on the dual spaces of simple
superalgebras with a non-degenerate Killing metric. The su(3,1) superalgebra is
given as a generic example.
