Based on an equivalent single conductor (ESC) model incorporating surface roughness scattering, the effect of the surface roughness of underlying dielectric material on the performance of multi-layer graphene nanoribbon (MLGNR) interconnects with side contacts is investigated. The performance of MLGNR interconnects with different doping concentrations on the most common dielectrics (SiO2, BN and SiC) is compared with that of Cu counterpart in terms of the time delay and the bandwidth. Results show that the surface roughness of dielectric material can severely degrade the performance of MLGNR interconnects, hence, it must be taken in to account in design considerations. While MLGNR interconnects with perfect edges on SiC substrate outperform Cu counterpart both in terms of time delay and bandwidth, for other substrate materials the performance of MLGNR interconnects with respect to Cu counterpart depends on doping concentration and interconnect length.