How can particles account for the curvature of spacetime? 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. 


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*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.
 A: Gravitons are a result of direct application of perturbative quantization to General Relativity. Because both perturbative QFT and GR are concise and renowned physical theories which have been checked in numerous experiments, there's little doubt that gravitons show up when we do high-energy (compared to the LHC, but low-energy compared to Planck scale) experiments with gravity.
In perturbation theory, one is always interested in small perturbations of the field around the mean value given by the classical solution. GR is no exception; in the approximation of perturbative theory we model the spacetime metric as
$$ g_{\mu \nu} (x) = \eta_{\mu \nu} + h_{\mu \nu} (x) $$
for small $h$, and work in powers of $h$. Note that this breaks background independence and the geometrical interpretation of GR, but we shouldn't expect an approximation to preserve it.
We know that perturbative quantum General Relativity is nonrenormalizable. From the Wilsonian point of view, this pretty much means that there's a domain of applicability beyond which it can no longer be trusted. This domain has always been identified with the Planck scale.
To summarize: perturbative quantum General Relativity is widely believed to be a valid description of the gravitational force in the low-energy regime, and it models the gravity field with particles called gravitons. Classical General Relativity is another approximation, which models the same field in a slightly different regime (strong but classical gravitational fields).
If any, gravitons play an interesting role in most of the quantum gravity models that we have today: they serve as a kind of sanity check. A sensible quantum gravity theory should give General Relativity in the classical limit and perturbative GR with gravitons in the limit of small geometry fluctuations. Both String Theory and Loop Quantum Gravity give gravitons in some limit.
