To the aim of studying the homogenization of low-dimensional periodic structures, we identify each of them to a periodic positive measure $\mu$ on $R^n$. We introduce a new notion of two-scale convergence for a sequence of functions $v_\epsilon \in L ^p_{\mu _\epsilon} (\Omega; R ^d)$, where $\Omega$ is an open bounded subset of $R^n$, and the measures $\mu _\epsilon$ are the $\epsilon$-scalings of $\mu$, namely, $\mu_\epsilon (B) := \epsilon ^n \mu (\epsilon ^ {-1}B)$. Enforcing the concept of tangential calculus with respect to measures, and related periodic Sobolev spaces, we prove a structure theorem for all the possible two-scale limits reached by the sequences $(u_\epsilon, \nabla u _\epsilon)$, when $\{u _\epsilon\} \subset {\cal C} ^1_0 (\Omega)$ satisfy the boundedness condition $\sup _\epsilon \int _{\Omega} |u_\epsilon| ^p + |\nabla \u_\epsilon| ^p \, d \mu_\epsilon < + \infty$ and when the measure $\mu$ satisfies suitable connectedness properties. This leads to deduce the homogenized density of a sequence of energies of the form $\int _{\Omega} j ({x \over \epsilon}, \nabla u) \, d \mu_\epsilon$, where $j(y,z)$ is a convex integrand, periodic in $y$, and satisfying a $p$-growth condition. The case of two parameter integrals is also investigated, in particular for what concerns the commutativity of the limit process.