EDIT: fixed thought experiment.
If antimatter anti-gravitates, it would be possible to build a perpetual motion machine:
1) Start with a zero-net momentum pair of photons. Have them collide, and make a particle/antiparticle pair. These can be moved around arbitrarily with zero net work, since any gravitational force on the particle will be equal and opposite to the force on the antiparticle.
2) raise them to an arbitrary height. Let them collide, producing a massive particle that is its own antiparticle, say a Higgs.
3) let this particle fall. It gains energy
4) let the particle decay upon falling. it now has the rest energy of the particle/antiparticle pair (equal to the energy of the two photons), plus any gravitational potential from falling.
5) reflect the two photons that are the decay product on antipodal mirrors attached by a wire and suspended from the ceiling. No net momentum or energy is transferred to this mirror system, and the photons can be freely merged again to repeat the process.