Right.. First off Brian Greene is one of my proffessors, and a very smart man. Unfortunately, dark energy and dark matter are FACTS, that won't change now, in our lifetime or ever. The probability that our experiments were wrong is something like 1 in 1000 at this stage. Whether or not our models are wrong, is irrelevant. Any new model of General relativity or whatever, will have to accomodate the data.
Two, Dark matter is not a problem for String theory, or any other current quantum gravity theory. Its more a problem for astronomers, and regular particle physicists. The current majority feel its the lightest supersymetric superpartner, which will be observed in about 4 years at CERN. If its not observed and/or doesnt exist, its not a big deal for quantum gravity theories, it just means whatever is causing the anomolous gravity is something else (there are many candidates).
Dark energy otoh, is a serious problem for our current understanding of physics and quantum gravity. Its really a big problem, for it is the state of absolute nothing that we don't understand. Current vacuum state calculations from quantum field theory gives an estimate that is either infinity, or with an additional likely assumption about 120 orders of magnitude off. I challenge anyone here to make a theory that gives a worse prediction.
Nearly everyone believes its the quantum gravity sector that is throwing things off.
String theory in its current form seems to give values for the cosmological constant (the simplest model for dark energy) that are negative, not positive. Everyone knows these are just toy models, and not realistic, however the general trend seems to indicate that things get 'negative' far easier than they get positive.
Since String theory is not really a 'theory' yet, but rather an ongoing program that studies quantum gravity, its unfair to dismiss it on this point.
When they get things ironed out mathematically, and parts start to get constrained and it still produces a bad prediction, then yes, its falsified.
You have to understand, in many ways, if you follow the logic of quantum field theory (the most successful theory of all times) according to String theorists, there is absolutely no alternative. All roads lead to string theory, is the popular saying. Its not exactly rigorous, but you can for instance change a whole slew of assumptions, and think your changing much of the dynamics of your theory. Yet in the end, almost miraculously, you end right back with String theory in disguise.
Thats why its been worked on by the smartest people on the planet for twenty years, despite formidable mathematical and conceptual barriers. Lines of problem solving that ask questions like 'well what else could it be?' all point strongly in favor of it.