While the energetic stability of the large number of possible SiO2 polymorphs has been widely addressed by both experimental and theoretical studies, there is a real dearth of information on their mechanical properties. We performed a systematic study of the elastic properties of 121 pure silica zeolites, including both experimentally synthesized and hypothetical structures, by means of density-functional theory calculations. We found that most frameworks exhibit high elastic anisotropy, and the experimentally synthesized structures are among the most mechanically stable ones. We propose to extend the "feasibility" criterion proposed in earlier literature to include elastic anisotropy, allowing to further reduce the number of possible targets for synthesis of SiO2 polymorphs. We also predict that a small number of hypothetical pure silica zeolites present large negative linear compressibility (NLC), which we attribute to the wine-rack motif of their framework.