The determination of the attenuation constants of periodic leaky-wave structures via the finite-difference time-domain (FDTD) method has been pursued so far via the simulation of a number of unit cells that is large enough to guarantee the convergence of the computed value. On the other hand, Brillouin diagrams of periodic structures can be readily extracted via the simulation of a single unit cell, terminated in periodic boundary conditions. This paper demonstrates a methodology that enables the concurrent extraction of leaky-wave attenuation constants and Brillouin diagrams of periodic structures, through the FDTD simulation of a unit cell. The proposed methodology is first validated and then employed to model leaky-wave radiation from a two-dimensional negative-refractive-index transmission-line (NRI-TL) medium. Apart from evaluating the characteristics of forward and backward leaky-wave radiation from such a medium, a lumped-element macro-model, with element values determined from the FDTD simulation, is extracted. The FDTD analysis, combined with this equivalent circuit, is used to investigate theoretically the possibility of the NRI-TL medium, as a leaky-wave antenna, to achieve continuous scanning from backward to forward end-fire and broadside radiation.