Periodic leaky-wave antennas are typically affected by the open-stopband (OSB) problem, i.e., the pattern degradation as the beam scans through broadside. Over the years, various techniques have been proposed to mitigate, or even suppress, this longstanding issue. However, most of these techniques cannot be easily applied to 1-D periodic 2-D structure. Recently, it has been theoretically and experimentally shown that the OSB can be suppressed by means of a very simple technique: it consists of designing a unit cell with two unequal discontinuities suitably spaced one another. In this work, we apply this novel technique to a canonical 1-D periodic 2-D structure, namely a grounded dielectric slab with a metal strip grating on top. To validate the concept, full-wave results are presented in the 17.5–20.5 GHz band considering an asymmetric unit-cell characterized by two narrow strips for the TM-polarized case.