  We consider proper, algebraic semismall maps f from a complex algebraic
manifold X. We show that the topological Decomposition Theorem implies a
"motivic" decomposition theorem for the rational algebraic cycles of X and, in
the case X is compact, for the Chow motive of X.The result is a Chow-theoretic
analogue of Borho-MacPherson's observation concerning the cohomology of the
fibers and their relation to the relevant strata for f. Under suitable
assumptions on the stratification, we prove an explicit version of the motivic
decomposition theorem. The assumptions are fulfilled in many cases of interest,
e.g. in connection with resolutions of orbifolds and of some configuration
spaces. We compute the Chow motives and groups in some of these cases, e.g. the
nested Hilbert schemes of points of a surface. In an appendix with T.
Mochizuki, we do the same for the parabolic Hilbert scheme of points on a
surface. The results above hold for mixed Hodge structures and explain, in some
cases, the equality between orbifold Betti/Hodge numbers and ordinary
Betti/Hodge numbers for the crepant semismall resolutions in terms of the
existence of a natural map of mixed Hodge structures. Most results hold over an
algebraically closed field and in the Kaehler context.
