The cubic temperature dependence of the nuclear relaxation rate (NRR) in layered superconductors with the order parameter having zeros at the Fermi surface (FS) is found to be universal under quite general conditions. The coefficient in the quasi-Korringa term for the NRR appearing at low temperatures due to impurity scattering is estimated. It is shown that an anisotropy of the gap function over the FS leads to the disappearance of the Hebel-Slichter coherence peak close to Tc. The corresponding analysis of the widely accepted model [D. Thelen et al., Phys. Rev. B 47, 9151 (1993)] for the Y-Ba-Cu-O system is presented.