(At the suggestion of the user markovchain, I have decided to take a very large edit/addition to the original question, and ask it as a separate question altogether.)

Here it is:

I have since thought about this more, and I have come up with an extension to the original question. The answers already given have convinced me that we can't just leave the metric as it is in GR untouched, but at the same time, I'm not convinced we have to quantize the metric in the way that the other forces have been quantized. In some sense, gravity isn't a force like the other three are, and so to treat them all on the same footing seems a bit strange to me. For example, how do we know something like non-commutative geometry cannot be used to construct a quantum theory of gravity. Quantum field theory on curved non-commutative space-time? Is this also a dead end?

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    $\begingroup$ It's true that "quantizing the metric in the way that the other forces have been quantized" fails. Goroff and Sagnotti proved the perturbative nonrenormalizability of GR. As far as I know (which isn't very far!) string theory seems to be the only game in town which treats gravity and the other forces on a similar footing. $\endgroup$ – twistor59 Feb 27 '13 at 17:57
  • $\begingroup$ @twistor59: you could plausibly argue that LQG treats gravity on an equal footing with the other forces, though one of the main issues with LQG is figuring out how to couple the other interactions to the theory. $\endgroup$ – Jerry Schirmer Feb 27 '13 at 18:35
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    $\begingroup$ You could also plausibly argue that LQG doesn't have GR as a low energy limit, since it seems to get the entropy of black holes wrong. See Sen's preprint arxiv.org/abs/1205.0971 $\endgroup$ – user1504 Feb 27 '13 at 18:38
  • $\begingroup$ @user1504: or that there's a flaw with Euclidean gravity analysis. LQG people would agree that LQG has a $\propto \hbar^{2}$ correction to the Hawking formula, which seems to be Sen's result from my fast reading of the abstract and the discussion beginning on page 27 of the preprint. $\endgroup$ – Jerry Schirmer Feb 27 '13 at 18:50
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    $\begingroup$ @JerrySchirmer I'd be interested to learn a bit about how that works, hence this question $\endgroup$ – twistor59 Feb 27 '13 at 20:22

Quantum field theory on curved non-communative spacetime is exactly what is used in the "semiclassical gravity" approach, an early example of which is Hawking's original derivation of the Hawking radiation effect. The limitation of this approach should be obvious--when the Hawking radiation ends up having a mass comparable to the original black hole, how can you trust the result? The Hawking radiation has mass and energy, too, and obviously, this should be factored into the result, but the semiclassical problem explicitly ignores it. (and if you try to do an iterative approach, factoring in the Hawking radiation as a source, and calculating the result, you quickly run into a LOT of complexity)

Non-communative geometry is an active area of research and a potential solution, albeit one chosen by a minority of researchers

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