  We study the motion of a classical spinning particle (with spin degrees of
freedom described by a vector of Grassmann variables) in higher-dimensional
general rotating black hole spacetimes with a cosmological constant. In all
dimensions n we exhibit n bosonic functionally independent integrals of
spinning particle motion, corresponding to explicit and hidden symmetries
generated from the principal conformal Killing--Yano tensor. Moreover, we
demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such
integrals are in involution, proving the bosonic part of the motion integrable.
We conjecture that the same conclusion remains valid in all higher dimensions.
Our result generalizes the result of Page et. al. [hep-th/0611083] on complete
integrability of geodesic motion in these spacetimes.
