Periodic binary systems can create complete acoustic bandgaps (i.e., stop bands) within which sound and vibrations are forbidden. This is of interest for applications such as acoustic filters, noise control, and improvements in transducers; as well as for pure physics concerned with the Anderson localization of sound and vibrations. The band structures for 3-D periodic arrays of water balloons surrrounded by mercury have been computed. Complete, multiple stop bands are found for fcc, bcc, and sc lattices. It is stressed that such a simple 3-D inhomogeneous system exhibits the widest stop bands ever reported for elastic as well as for dielectric composites. The gap/mid-gap ratios are found to be as high as ~1. For mercury balloons surrounded by water the gaps are found to be comparatively much smaller. Physical importance and applications of such gaps will be discussed.