We present the first experimental investigation of modulational instability in a layered Kerr medium. The particularly interesting and appealing feature of our configuration, consisting of alternating glass-air layers, is the piecewise-constant nature of the material properties, which allows a theoretical linear stability analysis leading to a Kronig-Penney equation whose forbidden bands correspond to the modulationally unstable regimes. We find very good quantitative agreement between theoretical, numerical, and experimental diagnostics of the modulational instability. Because of the periodicity in the evolution variable arising from the layered medium, there are multiple instability regions rather than just one as in a uniform medium.