The multidimensional Manhattan street networks constitute a family of digraphswith many interesting properties, such as vertex symmetry (in fact they are Cayleydigraphs), easy routing, Hamiltonicity, and modular structure. From the knownstructural properties of these digraphs, we determine their spectra, which alwayscontain the spectra of hypercubes. In particular, in the standard (two-dimensional)case it is shown that their line digraph structure imposes the presence of the zeroeigenvalue with a large multiplicity.