In this paper, we prove a weighted norm inequality for the Marcinkiewicz integral operator $mathcal{M}_{{Omega},h}$ when $h$ satisfies a mild regularity condition and ${Omega}$ belongs to $L(log L)^{1l2}(S^{n-1})$, $n{geq}2$. We also prove the weighted $L^p$ boundedness for a class of Marcinkiewicz integral operators $mathcal{M}^*_{{Omega},h,{lambda}}$ and $mathcal{M}_{{Omega},h,S}$ related to the Littlewood-Paley $g^*_{lambda}$-function and the area integral S, respectively.