The presence of wave tunneling and the “growing evanescent envelope” for field distributions in suitably designed, periodically layered stacks of frequency selective surfaces (FSS) is discussed in this letter. It is known that a slab of double-negative (DNG) or left-handed metamaterial sandwiched between two vacuum half-spaces may lead to “growing” evanescent plane waves, which may be used to restore sub-wavelength information in Pendry’s “perfect” lens (Phys. Rev. Lett., vol. 85, pg. 3966, 2000). In this letter, it is shown that a completely different setup allows an analogous buildup of evanescently modulated waves. In particular, it is shown how an interface resonance phenomenon similar to the one present at the interface between metamaterials with oppositely signed constitutive parameters may be induced by a proper choice of the periodicities of the FSS stacks and the geometrical properties of these surfaces. The analysis is performed through an equivalent transmission-line approach, and some physical insights into this phenomenon are presented. Salient features, such as the complete wave tunneling through the pair of cascaded FSS, each operating at its bandgap, are presented and discussed.