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In this work, the dispersion and radiation properties of bound and leaky modes on one-dimensional printed periodic structures (printed on a grounded dielectric slab) are investigated. It is shown that a new type of Brillouin diagram can be constructed to account for the different types of physical leakage, namely leakage into one or more surface waves or also simultaneously into space. This new Brillouin diagram provides physical insight into the behavior of the modes on one-dimensional periodic structures, and it also allows for a simple and convenient way to determine the paths of integration for the space harmonics, which arise in a spectral-domain moment-method solution of such structures. Numerical results are provided that show the advantage of using the new Brillouin diagram in explaining the leakage and stopband behavior for these kinds of periodic structures.