Would a directional "graviton" emitter violate any known laws of physics? Setting aside that we don't known what the mediating particle in quantum gravity looks like and have no way to manipulate it, what would the implications of a directional graviton source be? Would it allow a "reaction-less" drive without creating other problems (e.g. that a violation of conservation of momentum has corresponding implications with respect to translational invariance)?
Also, presuming that such a device (unlike mass) can be made to turn on and off, would that imply (via conservation of momentum and the finite propagation rate of gravity), that a graviton must have (negative?) momentum?

By "directional", I'm referring to a gravitational effect that, on average and over an arbitrarily long time, will attract particles more strongly in one hemisphere than the other. 
 A: Beyond the answer depending on a quantum theory of gravity, there is an important difference that need to be made. I will take QED as an example to express this point, as directional photon emitter do exists. In this simple analogy, mass would be charge and photons, gravitons. The point I want to illustrate is related to the so-called "virtual" particles.
When two charged particles interact, this interaction is often pictured as relying on the exchange of virtual photons. These corresponds to the inner edges of the Feynman diagrams associated to the interaction. These virtual particles are best seen, in my humble opinion, as possible interaction channels. This way, we see the two charged particle as interacting via the photon field. There can also be freely propagating photons. These are the ones that are associated to the external edges of Feynman diagrams. 
Likewise, for the case of gravity, we would expect that the gravitational interaction of two masses is mediated via the exchange of "virtual" gravitons. I suppose these are not the type you are referring about since they are not freely propagating. Indeed, freely propagating gravitons would be more related to gravitational waves. A classical gravitational wave would be expected to be some kind of superposition of gravitons, much the same way classical electromagnetic waves can be seen as a superposition of photons.
Pushing the analogy, lasers are directional sources of electromagnetic waves and the photons in the beam are in a coherent state. I would expect your directional graviton source to be some kind of "gravitational wave laser". In fact, this is something that probably could exist in principle.
Now, regarding your point on a "reaction-less" drive, a "gravity laser" would not be properly such a drive. If what you mean by reaction-less drive is just a drive that would not require propellant mass, than these already exists in the form of directional light sources. No violation of physical principles is involved as photons can carry momentum. Likewise, it may be possible to base a drive on a directional gravitational wave source, but I would not see the point, because there would be no inherent advantage to using a directional light source (since both gravitons and photons are massless, there would be no increase in the energy efficiency), but the complexity of the device would probably be orders of magnitude bigger as user dmckee suggested in his comments to your question.
Moreover, this would suggest that the gravitons must have negative momentum only if you define positive momentum in the direction the spaceship using such a drive is accelerating. In general, gravitons would not have "negative momentum" as this is essentially meaningless, the momentum being a vector quantity the sign of which depend on an inconsequent choice of space orientation if at all defined.
If you rather meant "negative momentum" as meaning that the if the beam emitted by your drive now collides with matter, it will act as an attractive force, then this is certainly not the case, as this would imply a violation of momentum conservation, as you suggested. In fact, the gravitational waves would probably diffract on the mass it "collides" with. A way to phrase it would be to say that some part of the wave would go right through and another part would be absorbed. The part that is absorbed could then be partly re-emitted. At the end the fraction of the momentum of the wave that is absorbed would be transferred to the mass. On the level of freely propagating gravitons, I suggest this would depend on the cross section of the graviton with the mass. With some probability, it will not interact and go right through and with some other probability it will be absorbed and maybe re-emitted, transferring momentum to the mass it collides with so that the total momentum is conserved, overall having a repulsive or no effect at all. An attractive force would rather involve virtual gravitons, unlike the one in the output of your device.
Finally, concerning your last comment, I am not sure of what you meant. It seemed you confused the "virtual" gravitons with the freely propagating ones. If particles are attracted more by one hemisphere, it would mean there is more mass in that hemisphere. It wouldn't have anything to do with the directionality of the emission of the freely propagating gravitons.
A: This is what happens when a gravitational wave passes . It does come from the direction of the source, but note the classical framework of general relativity.
Gravitons are the province of quantum mechanics and one can only speculate , until a definitive quantization of gravity is obtained.
Nevertheless, it is known that the classical wave will be composed by innumerable gravitons, the way classical electromagnetic waves emerge from innumerable photons. One can safely predict that conservation laws in the flat space of a limited environment that you imply all conservation laws will hold and there will only be a negative ! momentum if the coordinate system is designed that way. A machine emitting gravitons will have equal and opposite momentum to the gravitons leaving it.
One can make gravitational quadrupoles easily but generating enough energy is the problem, with the very weak couplings of gravity it is only with LIGO they  detected gravitational waves, and individual gravitons have not been seen , due to the very small coupling constant.
So even if in principle one could design a graviton source, one needs astrophysical dimensions to be able to get enough energy in the wave generated .
