This paper presents a method to enhance the light absorption and power conversion efficiency of organic solar cells (OSCs) by embedding a plasmonic Ag nanocuboid array into the active layer. Numerical simulations based on the finite-difference time-domain method are conducted to compare the enhancement of short circuit current density (Jsc) caused by the Ag nanocuboid array with other types of plasmonic nanostructures such as nanocubes, nanospheres, nanorods, and nanocylinders. It is demonstrated that the nanocuboid array can lead to an enhancement of 25.5% in Jsc, much higher than that of other nanostructure arrays. Analyses of the photoelectric field and light absorption enhancement show that the enhancement in Jsc primarily results from the combined effects of localized surface plasmon resonance (LSPR) and optical interference within the devices. It is also revealed that the optical interference can improve or weaken the absorption-enhancing ability of the LSPR mode, which depends on the spectral position of the LSPR mode and the spatial position of the Ag nanocuboids in the optical electric field. Finally, we investigate the effects of light polarization and nanostructure size on Jsc enhancement of the OSC devices. The findings in the paper provide theoretical support for designing OSCs with thinner active layers and superior absorption performance.