We demonstrate with the finite-difference time-domain method that radiative heat transfer between two parallel gold plates can be significantly enhanced by engraving periodic grooves with a subwavelength width on the plate surfaces. The enhancement increases with a decrease in the separation distance at near-field regime and it can be further efficiently improved by having a supercell with multiple grooves with different depths. We attribute this near-field enhancement to coupling of thermally excited spoof surface plasmon polaritons, a type of artificial surface wave inherent to structured metal surfaces [J. B. Pendry, L. Mart´ın-Moreno, and F. J. Garcia-Vidal, Science 305, 847 (2004)]. The frequency-dependent contribution to the heat transfer, or transmission-factor spectrum, is confirmed by calculating the dispersion relation of guided modes by the two parallel corrugated plates through a finite-element method. Especially, the photonic density of states derived from the dispersion relation is found to have excellent agreement to the transmission-factor spectrum.