Convolution type Calderón-Zygmund singular integral operators with rough homogeneous kernels p.v. \(\Omega(x)/| x|^n\) are studied. It is proved that if \(\Omega\) satisfies \[ \sup_{\xi\in S^{n- 1}} \int_{S^{n-1}}|\Omega(\theta)| \Biggl(\ln{1\over|\theta\cdot \xi|}\Biggr)^{1+\alpha} d\theta< \infty\tag{1} \] then the corresponding singular integrals and maximal singular integrals map \(L^p\to L^p\) for \(1<p<2+ \alpha\) and \(1< p<2(2+ \alpha)/3\), correspondingly. As a corollary, if condition (1) is satisfied for every \(\alpha> 0\), then \(T_\Omega, T^*_\Omega: L^p\to L^p\) for \(1<p<\infty\). This condition is shown to be different from the condition \(\Omega\in H^1(S^{n-1})\), which was the best known.