We theoretically investigate a new class of silicon waveguides for achieving Stimulated Brillouin Scattering (SBS) in the mid-infrared (MIR). The waveguide consists of a rectangular core supporting a low-loss optical mode, suspended in air by a series of transverse ribs. The ribs are patterned to form a finite quasi-one-dimensional phononic crystal, with the complete stopband suppressing the transverse leakage of acoustic waves, confining them to the core of the waveguide. We derive a theoretical formalism that can be used to compute the opto-acoustic interaction in such periodic structures, and find forward intramodal-SBS gains up to 1750 m-1W-1, which compares favorably with the proposed MIR SBS designs based on buried germanium waveguides. This large gain is achieved thanks to the nearly complete suppression of acoustic radiative losses.