The generalized multiparticle Mie-solution (GMM), a Lorenz-Mie-type rigorous theory for the scattering of a monochromatic plane wave by an arbitrary configuration of nonintersecting scattering bodies, has lately been revisited and further developed. A recent progress is the initiation of a special version applied to one- and two-dimensional (1D and 2D) periodic arrays (PAs) of identical particles [J. Opt. Soc. Am. A30, 1053 (2013)]. As a continuous advance, the present work extends the initiative PA-type solution from 1D and 2D to the more involved three-dimensional (3D) regular arrays. Analytical formulations applicable to the 3D PAs are derived, including the special PA-type explicit expressions for cross sections of extinction, scattering, backscattering, and radiation pressure. The specific PA-version is a complement to the general formulation and solution process of the standard GMM. In either 1D and 2D or 3D cases, the newly devised PA-approach is capable of providing expeditiously theoretical predictions of radiative scattering characteristics for periodic structures consisting of a huge number of identical unit cells, which the general approach of the GMM is unable to handle in practical calculations, owing to excessive computing time and/or computer memory requirements. To illustrate practical applications, sample numerical solutions obtained via the PA-approach are shown for 3D PAs of finite lengths that have ∼5×10⁷ component particles, including structures having a rectangular opening. Also discussed is potential future work on the theory and its tests.