Classical General Relativity rests on the idea that what we call gravity actually is one property of spacetime itself. The matter distribution determines the metric by means of the Einstein field equations and hence the associated Levi-Civita connection which tells how particles move on such a background.
Now, classical electrodynamics is different. We picture spacetime as already there, and then we picture the electromagnetic field as something "on top" of the background spacetime.
Thus the transition to QED, where electromagnetic interaction ends up becoming exchange of photons, is not that strange. Instead of having a field value on each event, we have exchange of photons all the time on that spacetime background.
Now, it seems quite well accepted that a quantum theory of gravity will have one corresponding graviton, which I believe will mediate the gravitational force by means of exchange of such particles.
But things are different in gravity. Gravity is not a force, it is a property of spacetime.
How can gravity be described by exchange of particles if it is a property of spacetime and not a force?
How can this exchange of particles even take place, if there is no background in this case? I believe that gravity and spacetime are somehow inseparable, i.e., if there is no classical gravity, no spacetime.
How can the geometry of spacetime (the manifold with the metric and connection) be accounted for with exchange of particles?
Since the graviton idea seems quite well accepted (I've seem people say that theoretically it is fine, it just hasn't been detected), I believe these questions have plausible answers at least.