Several experiments have shown a huge enhancement in thermal radiation over the black-body limit when two objects are separated by nanoscale gaps. Although those measurements only demonstrated enhanced radiation between homogeneous materials, theoretical studies now focus on controlling the near-field radiation by tuning surface polaritons supported in nanomaterials. Here, we experimentally demonstrate near-field thermal radiation between metallo-dielectric multilayers at nanoscale gaps. Significant enhancement in heat transfer is achieved due to the coupling of surface plasmon polaritons (SPPs) supported at multiple metal-dielectric interfaces. This enables the metallo-dielectric multilayers at a 160-nm vacuum gap to have the same heat transfer rate as that between semi-infinite metal surfaces separated by only 75 nm. We also demonstrate that near-field thermal radiation can be readily tuned by modifying the resonance condition of coupled SPPs. This study will provide a new direction for exploiting surface-polariton-mediated near-field thermal radiation between planar structures.