Several experiments have shown a huge enhancement in thermal radiation over the blackbody 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.