Let X be a complete intersection inside a variety M with finite-dimensional motive and for which the Lefschetz-type conjecture B(M) holds. We show how conditions on the niveau filtration on the homology of X influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if M has trivial Chow groups and if X has non-trivial variable cohomology parametrized by c-dimensional algebraic cycles, then the cycle class maps A k (X) → H 2k (X) are injective for k<c. We give variants involving group actions, which lead to several new examples with finite-dimensional Chow motives.