Mathematical models for fluid transport, elasticity, and wave propagation in heterogeneous periodic media often rely on partial differential equations completed with periodic boundary conditions. An accurate numerical approximation of such problems requires a spatial discretization that satisfies the same periodicity constraints, in order to prevent spurious discretization effects. However, constructing periodic computational meshes is not trivial, especially in three dimensions, where vertices, edges, and faces on the periodic boundary must match consistently. While ad hoc meshes can be built by carefully controlling boundary entities, such workflows are difficult to automate, are typically limited to isotropic tessellations, and become impractical in simulations where meshes have to be modified through adaptive routines. In this presentation, we introduce an anisotropic metric-based mesh adaptation algorithm for periodic 3D domains, focusing on tetrahedral tessellations for finite element simulations. The proposed procedure enforces periodicity throughout the adaptation loop via minimal, localized operations, so that mesh entities on paired boundaries match exactly. The algorithm is tested on several cases to assess adaptation performance under different periodic requirements and anisotropy regimes. The results show that the proposed procedure is fully automatic and independent of the geometry, the periodic mapping, and the adaptation strategy. In addition to benchmark numerical experiments, we illustrate the impact of the proposed algorithm on engineering applications by coupling periodic mesh adaptation with inverse homogenization topology optimization for metamaterial design.