Recently, massive early-type galaxies have shed their red-and-dead moniker, thanks to the discovery that many host residual star formation. As part of the ATLAS-3D project, we have conducted a complete, volume-limited survey of the molecular gas in 260 local early-type galaxies with the IRAM-30m telescope and the CARMA interferometer, in an attempt to understand the fuel powering this star formation. We find that around 22% of early-type galaxies in the local volume host molecular gas reservoirs. This detection rate is independent of galaxy luminosity and environment. Here we focus on how kinematic misalignment measurements and gas-to-dust ratios can be used to put constraints on the origin of the cold ISM in these systems. The origin of the cold ISM seems to depend strongly on environment, with misaligned, dust poor gas (indicative of externally acquired material) being common in the field but completely absent in rich groups and in the Virgo cluster. Very massive galaxies also appear to be devoid of accreted gas. This suggests that in the field mergers and/or cold gas accretion dominate the gas supply, while in clusters internal secular processes become more important. This implies that environment has a strong impact on the cold gas properties of ETGs.