We demonstrate three-dimensional surface-enhanced Raman spectroscopy (SERS) substrates formed by accumulating plasmonic nanostructures that are synthesized using a DNA-assisted assembly method. We densely immobilize Au nanoparticles (AuNPs) on polymer beads to form core-satellite nanostructures for detecting molecules by SERS. The experimental parameters affecting the AuNP immobilization, including salt concentration and the number ratio of the AuNPs to the polymer beads, are tested to achieve a high density of the immobilized AuNPs. To create electromagnetic hot spots for sensitive SERS sensing, we add a Ag shell to the AuNPs to reduce the interparticle distance further, and we carefully adjust the thickness of the shell to optimize the SERS effects. In addition, to obtain sensitive and reproducible SERS results, instead of using the core-satellite nanostructures dispersed in solution directly, we prepare SERS substrates consisting of closely packed nanostructures by drying nanostructure-containing droplets on hydrophobic surfaces. The densely distributed small and well-controlled nanogaps on the accumulated nanostructures function as three-dimensional SERS hot spots. Our results show that the SERS spectra obtained using the substrates are much stronger and more reproducible than that obtained using the nanostructures dispersed in solution. Sensitive detection of melamine and sodium thiocyanate (NaSCN) are achieved using the SERS substrates.