We report the first realization of a three-dimensional (3D) acoustic double-zero-index medium (DZIM) made of a cubic lattice of metal rods. While the past decade has seen several realizations of 2D DZIM, achieving such a medium in 3D has remained an elusive challenge. Here, we show how a fourfold degenerate point with conical dispersion can be induced at the Brillouin zone center, such that the material becomes a 3D DZIM with the effective mass density and compressibility simultaneously acquiring near-zero values. To demonstrate the functionalities of this new medium, we have fabricated an acoustic waveguide of 3D DZIM in form of a "periscope" with two 90° turns and observed tunneling of a normally incident planar wave through the waveguide yielding undistorted planar wave front at the waveguide exit. Our findings establish a practical route to realize 3D DZIM as an effective acoustic "void space" that offers unprecedented control over acoustic wave propagation.