Architected materials derive their exceptional mechanical performances from deliberately designed micro- and meso-scale configurations rather than from their base material properties alone. By exploiting ad hoc programmed internal arrangement of the constituents, local and/or global instability mechanisms, and concealed pre-stress distributions, these systems in fact allow to attain unconventional and tuneable macroscopic responses, difficult to observe in classical continua. In this contribution, we present an analytical multi-scale modelling framework aimed at highlighting how micro-structural design can govern the emergent mechanical behaviour of periodically organized constructs. In particular, the transition from discrete architectures to effective continuum descriptions is addressed within the theory of Structured Deformations, thus providing a novel homogenization strategy capable of capturing elastic instabilities and kinematical disarrangements occurring at the micro-scale and to project them to the macroscopic level. The analysis begins with a one-dimensional multi-element system, which serves as a paradigmatic model to show how a wide class of nonlinear hyperelastic constitutive behaviours can be naturally derived by tailoring the tensile buckling response at the discrete scale. The study is then extended to two-dimensional configurations conceived for obtaining enriched mechanical properties, such as the occurrence of global auxeticity from an initially symmetric and compact medium and tensile-induced elastic instabilities in a ring-like system. The torsional response of a structured cylinder is finally analyzed, revealing asymmetric, non-monotonic, and tuneable Poynting effect, governed by the geometrical arrangement of the micro-structure. Overall, the proposed approach provides predictive capabilities for the effective behaviour of structured continua and meta-materials, and establishes an analytical basis for their parametric design and optimization, with potential applications across several engineering domains.