Nature of gravity: gravitons, curvature of space-time or both?

Well, consider this: the same thing happens with electromagnetic forces. We can describe them as particles responding to the presence of electric and magnetic fields, or we can describe them as resulting from the exchange of virtual photons. Those views seem similarly incompatible, but nevertheless both theories (classical electrodynamics and quantum electrodynamics, respectively) give excellent predictions. We can't really say that one is more "right" than another; we just have to accept them both.

The situation with gravity is pretty much a direct analogy to electromagnetism. We can describe gravity as particles responding to the presence of spacetime curvature, or we can describe them as resulting from the exchange of virtual gravitons. As with EM, these views would correspond to classical gravity and quantum gravity, respectively. But the difference is that, although general relativity fills the role of the classical theory, we don't have a good quantum theory of gravity yet.

I wouldn't say that the field/particle duality is one of the problems that impedes the combination of quantum mechanics with GR. After all, we had no problem getting around the dual descriptions of electromagnetism. It's just the peculiar details of quantum gravity that make it a difficult theory to develop.

Here is just a small remark. It is possible to give a strict mathematical proof about the equivalence of these two pictures.

If you just start with the three (semi-experimental) facts: Lorentz invariance, $1/r$ long-range tail of gravitational force and its one-way action (attraction only) and the fact that the bending of light almost doesn't depend on its frequency and polarization, then you will find that these facts are compatible (in the large distance limit) only with the massless helicity $\pm2$ particle exchange. After that, it has been proved that special relativity and analytic properties of scattering amplitude lead to the equivalence principle [1,2]. This theorem is a pure analog of Gell-Mann-Low-Goldberger soft photon theorem, which claims that the power expansion of the amplitude of photon scattering by a hadron (with respect to photon frequency) does not depend on the spin or internal structure of the hadron (up to the second order). By considering multigraviton scattering amplitudes one can prove that the all local vertices for soft gravitons correspond to the expansion of the Einstein action.

It means that the exchange of helicity $\pm2$ massless particle unavoidably leads to the classical general relativity (the opposite statement is trivial).

This program was initiated by Steven Weinberg [1,2] and finished by Deser and Boulware [3]. You can find the complete consideration in their paper [3] with the title “Classical general relativity derived form quantum gravity”. This paper is a real masterpiece of clear physical explanation of this problem.

References

[1] S. Weinberg, Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of gravitational and inertial mass, Phys. Rev. B135 (1964) 1049.

[2] S. Weinberg, Photons and gravitons in perturbation theory: derivation of Maxwell’s and Einstein’s equations, Phys. Rev. B138 (1965) 988.

[3] D. G. Boulware, S. Deser, Classical general relativity derived form quantum gravity, Ann. Phys. 89 (1975) 193.

Although the analogy between gravity and electromagnetism made by David is fine and self-suggesting, it must be cautiously added that there is no proof that gravity must look like exchange of gravitons at the microscopic level. We actually do not know what the microscopic picture of gravity is, and it might turn out to be very different from the familiar description in terms of particle-carriers of that force.

For example, earlier this year there was a preprint by Erik Verlinde suggesting that gravity might be an entropic force. It this is true, gravitons do not appear in this picture at all. This preprint is being actively discussed (more that 100 citations this year). However it also must be said that Verlinde's suggestion still remains a suggestion, not a theory, as it relies on some murky heuristic arguments, not a solid mathematical theory.

Update: the commenter below correctly points out that regardless of the microscopic theory the large-wavelength gravitational waves exist in any case and they can be quantized giving rise to gravitons. So, I guess my caution was misleading.