  We make use of interferometric CO and HI observations, and optical
integral-field spectroscopy to probe the origin of the molecular and ionised
interstellar medium (ISM) in local early-type galaxies (ETGs). We find that
36\pm5% of our sample of fast rotating ETGs have their ionised gas
kinematically misaligned with respect to the stars, setting a strong lower
limit on the importance of externally acquired gas (e.g. from mergers and cold
accretion). Slow rotators have a flat distribution of misalignments, indicating
that the dominant source of gas is external. The molecular, ionised and atomic
gas in all the detected galaxies are always kinematically aligned, even when
they are misaligned from the stars, suggesting that all these three phases of
the ISM share a common origin. In addition, we find that the origin of the cold
and warm gas in fast-rotating ETGs is strongly affected by environment, despite
the molecular gas detection rate and mass fractions being fairly independent of
group/cluster membership. Galaxies in dense groups and the Virgo cluster nearly
always have their molecular gas kinematically aligned with the stellar
kinematics, consistent with a purely internal origin. In the field, however,
kinematic misalignments between the stellar and gaseous components indicate
that >46% of local fast-rotating ETGs have their gas supplied from external
sources. We discuss several scenarios which could explain the environmental
dichotomy, but find it difficult to simultaneously explain the kinematic
misalignment difference and the constant detection rate. Furthermore, our
results suggest that galaxy mass may be an important independent factor
associated with the origin of the gas, with the most massive fast-rotating
galaxies in our sample (M_K<-24 mag; stellar mass of >8x10^10 Msun) always
having kinematically aligned gas. (abridged)
