Much of the difficulty in merging these theories comes from the radically different assumptions that these theories have on how the universe works. Quantum mechanics depends on particle fields embedded in the flat space-time of either Newtonian mechanics or special relativity. Einstein's theory of general relativity models gravity as a curvature within space-time that changes as mass moves. The most obvious ways of combining the two (such as treating gravity as simply another particle field) run quickly into what is known as the renormalization problem. Gravity particles would attract each other and if you add together all of the interactions you end up with many infinite results which can not easily be cancelled out. This is in contrast with quantum electrodynamics where the interactions do result in some infinite results, but those are few enough in number to be removable via renormalization.
Another difficulty comes from the success of both quantum mechanics and general relativity. Both have been highly successful and there are no known phenomenon that contradict the two. The energies and conditions at which quantum gravity are likely to be important are inaccessible to laboratory experiments. The result of this is that there are no experimental observations which would provide any hints as to how to combine the two.
The general approach taken in deriving a theory of quantum gravity is to assume that the underlying theory will be simple and elegant and then to look at current theories for symmetries and hints for how to combine them elegantly into a overarching theory. One problem with this approach is that it is not known if quantum gravity will be a simple and elegant theory.
Such a theory is required in order to understand those problems involving the combination of very large mass or energy and very small dimensions of space, such as the behaviour of black holes, and the origin of the universe.
There are a number of proposed quantum gravity theories and proto-theories, including (for example) string theory and the loop quantum gravity of Smolin and Rovelli - see http://www.livingreviews.org/Articles/Volume1/1998-1rovelli/
The Noncommutative geometry of Alain Connes, and Twistor theory, of Roger Penrose, are also theories of quantum gravity
Quantum gravity theorists: