We report our spectroscopic investigation of the transiting ice giant GJ 3470b's atmospheric transmission, and the first results of extrasolar planet observations from the new Keck/MOSFIRE spectrograph. We measure a planet/star radius ratio of 0.0789in a bandpass from 2.09-2.36 µm and in six narrower bands across this wavelength range. When combined with existing broadband photometry, these measurements rule out cloud-free atmospheres in chemical equilibrium assuming either solar abundances (5.4σ confidence) or a moderate level of metal enrichment (50× solar abundances, 3.8σ), confirming previous results that such models are not representative for cool, low-mass, externally irradiated extrasolar planets. Current measurements are consistent with a flat transmission spectrum, which suggests that the atmosphere is explained by high-altitude clouds and haze, disequilibrium chemistry, unexpected abundance patterns, or the atmosphere is extremely metal-rich ( > ∼ 200× solar). Because GJ 3470b's low bulk density sets an upper limit on the planet's atmospheric enrichment of < ∼ 300× solar, the atmospheric mean molecular weight must be < ∼ 9. Thus, if the atmosphere is cloud-free its spectral features should be detectable with future observations. Transit observations at shorter wavelengths will provide the best opportunity to discriminate between plausible scenarios. We obtained optical spectroscopy with the GMOS spectrograph, but these observations exhibit large systematic uncertainties owing to thin, persistent cirrus conditions. Finally, we also provide the first detailed look at the steps necessary for well-calibrated MOSFIRE observations, and provide advice for future observations with this instrument.