We investigate scattering, localization and dispersive time-decay properties for the one-dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential, $q_\epsilon=q(x,x/\epsilon)$, where $q(x,y)$ is periodic and mean zero with respect to $y$. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low-energy ($k$ small) behavior of scattering quantities, e.g. the transmission coefficient, $t^{q_\epsilon}(k)$, as $\epsilon$ tends to zero. We derive an effective potential well, $\sigma_{\rm eff}(x)=-\epsilon^2\Lambda_{\rm eff}(x)$, such that $t^{q_\epsilon}(k)-t^{\sigma_{\rm eff}}(k)$ is small, uniformly for $k\in\mathbb{R}$ as well as in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled limit of the transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if $\epsilon$, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half plane, on the imaginary axis at a distance of order $\epsilon^2$ from zero. It follows that the Schrödinger operator $H_{q_\epsilon}=-\partial_x^2+q_\epsilon(x)$ has an $L^2$ bound state with negative energy situated a distance $\mathcal{O}(\epsilon^4)$ from the edge of the continuous spectrum. Finally, we use this detailed information to prove a local energy time-decay estimate.