Nonlinear waves on periodic backgrounds play an important role in physical systems. In this study, nonlinear waves that include solitons, breathers, rogue waves, and semi-rational solutions on periodic backgrounds for the coupled Lakshmanan-Porsezian-Daniel equations are investigated. Moreover, the interactions between different types of nonlinear waves are examined and their dynamic behaviors are studied. In particular, it is observed that bright-dark rogue waves interact with bright-dark breathers or solitons on periodic backgrounds, four-petaled breathers interact with two eye-shaped breathers on periodic backgrounds, and a four-petal rogue wave interplays with a rogue wave on periodic backgrounds. Furthermore, it is found that the value of the parameter 𝜸 3 affects the weak and strong interactions of these nonlinear waves. These results may be useful in the study of nonlinear wave dynamics in coupled nonlinear wave models.