We consider Marcinkiewicz integrals arising from rough kernels satisfying the $L\log L$ condition on the unit $(n-1)$-sphere and prove the weak type (1,1) estimates. We also prove the weighted weak type (1,1) estimates with certain $A_1$-weights. In this case the $L\log L$-condition is replaced by the $L^q$-condition with $q > 1$.
