We report on laboratory measurements of compressional and shear wave speeds in cylindrical, compacted, polycrystalline methane hydrate and ice Ih samples. The methane hydrate sample was made from granulated ice warmed to 290 K in a pressurized methane atmosphere. The resultant porous methane hydrate was compacted within the synthesis vessel using pistons containing custom designed piezo-ceramic transducers. During compaction and for the remainder of the experiment, compressional and shear waveforms generated by the transducers were digitally recorded for later determination of compressional and shear wave speeds in the sample. Once the methane hydrate sample was fully compacted, the temperature was twice cycled in steps from 258 to 288 K. At each temperature, the uniaxial pressure was held constant for three hours to insure the sample reached physical equilibrium. At the conclusion of the second temperature cycle, the temperature was held constant (288 K) and the piston pressure was decreased by 500 psi and then held constant for one hour. 500 psi pressure steps continued until the piston was pushed off the sample by methane gas pressure. The ice experiment was similar, but with the following differences: no methane gas was used, the sample was evacuated before compaction, the temperature range was 253 to 268 K and the piston pressure was slowly varied at each temperature instead of decreased in steps at the end of the second temperature cycle. Gas pressure, piston pressure, temperature and sample length were recorded throughout both the methane hydrate and ice experiments. Results from these experiments illustrate several relationships between ice Ih and methane hydrate: 1) Methane hydrate resists compaction much more than ice Ih. A piston pressure of 6,000 psi fully compacted the ice sample at 268 K. A pressure of 15,000 psi was required to fully compact the methane hydrate sample at 253 K. 2) Modeling wave speed increases at constant sample length strongly suggests grain to grain bonds form between adjacent ice or methane hydrate grains. The relative wave speed increases with time show that ice is much more efficient at this process, perhaps due to the higher mobility of water in ice's crystal lattice. 3) Linear regressions fit to the wave speed data show that within the pressure and temperature conditions of the experiments, compressional wave speed is greater in ice and shear wave speed is greater in methane hydrate. As a result, the wave speed based calculation of Poisson's ratio is larger in ice than it is in methane hydrate. 4) Shear wave speed decreases with increasing confining pressure in both ice and methane hydrate. This behavior is well known in ice Ih. No previous measurements of shear wave speed in methane hydrate versus pressure have been reported, so no other data are available for comparison. 5) Within the pressure and temperature ranges of the experiments, shear wave speeds in ice vary more with temperature and pressure than they do in methane hydrate and compressional wave speeds in ice vary more with temperature and less with pressure than they do in methane hydrate. Additional experiments are planned to confirm and extend these findings.