Simultaneous measurements of compressional and shear wave speeds through polycrystalline methane hydrate have been made. Methane hydrate, grown directly in a wave speed measurement chamber, was uniaxially compacted to a final porosity below 2%. At 277 K, the compacted material's compressional wave speed was 3650 { -} 50 m/s. The shear wave speed, measured simultaneously, was 1890 { -} 30 m/s. From these wave speed measurements, we derive Vp/Vs, Poisson's Ratio, bulk, shear and Young's moduli.