The propagation of a normally incident plane acoustic wave through a three-dimensional rigid slab with periodically placed holes is modeled and analyzed. The spacing of the holes A and B, the wavelength lambda, and the thickness of the slab L are order one parameters compared to the characteristic size D of the holes, which is a small quantity. Scattering matrix techniques are used to derive expressions for the transmission and reflection coefficients of the lowest mode. These expressions depend only on the transmission coefficient, tau(0), of an infinitely long slab with the same configuration. The determination of tau(0) requires the solution of an infinite set of algebraic equations. These equations are approximately solved by exploiting the small parameter D/square root(AB). Remarkably, this structure is transparent at certain frequencies and opaque for all others. Such a structure may be useful in constructing narrow-band filters and resonators.