Let mu be a non-negative Radon measure on R-d which satisfies only some growth conditions. Under this assumption, the boundedness in some Hardy-type spaces is established for a class of maximal Calderon-Zygmund operators and maximal commutators which are variants of the usual maximal commutators generated by Calderon-Zygmund operators and RBMO(mu) functions, where the Hardytype spaces are some appropriate subspaces, associated with the considered RBMO(mu) functions, of the Hardy space H-1(mu) of Tolsa.