We performed fluorescence correlation spectroscopy measurements to assess the long-time self-diffusion of a variety of spherical tracer particles in periodic porous nanostructures. Inverse opal structures with variable cavity sizes and openings in the nanometer domain were employed as the model system. We obtained both the exponent of the scaling relation between mean-square displacement and time and the slow-down factors due to the periodic confinement for a number of particle sizes and confining characteristics. In addition, we carried out Brownian dynamics simulations to model the experimental conditions. Good agreement between experimental and simulation results has been obtained regarding the slow-down factor. Fickian diffusion is predicted and seen in almost all experimental systems, while apparent non-Fickian exponents that show up for two strongly confined systems are attributed to polydispersity of the cavity openings. The utility of confining periodic porous nanostructures holds promise toward understanding of constrained diffusion with a wide range of applications ranging from water purification and drug delivery to tissue engineering.