We prove a weighted $L^p$ boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space $L^p({\mathbb{R}}^n\times {\mathbb{R}}^m,{\omega }_1(x)dx,{\omega }_2(y)dy)$, provided that the weights ${\omega }_1$ and ${\omega }_2$ are certain radial weights and that the kernels are rough in the optimal space $L(\log L)({\mathbb{S}}^{n-1}\times {\mathbb{S}}^{m-1})$. In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted $L^p$ boundedness of the related square and maximal functions. Our weighted $L^p$ inequalities extend as well as generalize previously known $L^p$ boundedness results.