Numerous methods to increase the stimulated Brillouin scattering (SBS) threshold have been previously implemented. Some are passive, based on acousto-optic fiber designs that incorporate longitudinally-or radially-tailored optical and/or acoustic index profiles, leading to broadened Brillouin gain spectra (BGS) with reduced peak gain. Some are active, relying on an applied temperature or strain distribution, also resulting in broadened BGS. Broadening the laser spectrum still represents the most effective method to-date to obtain large-scale (> 20 dB) decreases in the gain, but the suitability of this method depends largely on the application and system requirements on the laser spectrum. Despite these technologies, some introduced only in the last decade, the vast majority of high-energy, narrow-linewidth fiber laser systems are still limited by SBS rather than the availability of pump power. We present an alternative approach; rather than focusing on 'suppressing' SBS in waveguide or other designs, we propose implementing materials with intrinsically low Brillouin gain. We focus on high-density, high-sound-velocity, large acoustic-damping-coefficient, and low-photoelastic-constant materials wherein the correct balancing of physical characteristics gives rise to extremely low Brillouin gain. In general, the approach requires the use of compositions that would be considered to be highly unconventional and unachievable utilizing standard fiber fabrication methods. For example, we describe recent results on sapphire-derived fibers (among other compositions) wherein a Brillouin gain nearly 20 dB lower than those of more conventional fibers has been realized. Other compositions will also be presented, including new results on a novel baria doped fiber, including others predicted to have zero-valued photoelastic constants, and therefore zero Brillouin gain.