  We study links between algebraic cycles on threefolds and
finite-dimensionality of their motives with coefficients in Q. We decompose the
motive of a non-singular projective threefold X with representable algebraic
part of CH_0(X) into Lefschetz motives and the Picard motive of a certain
abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X)
when the ground field is C. In particular, it implies motivic
finite-dimensionality of Fano threefolds over a field. We also prove
representability of zero-cycles on several classes of threefolds fibered by
surfaces with algebraic H^2. This gives another new examples of
three-dimensional varieties whose motives are finite-dimensional.
