The trick is in the substrate and amount of material necessary to achieve any kind of functioning device:
Organic semiconductors can be processed at sufficiently low temperature so that you can use flexible substrates like plastic foils or paper, even when you're doing PVD of small molecules in vacuum. To a certain extent, this also works for oxide semiconductors, but not so much for Si or other conventional materials because of the high processing temperatures.
Additionally, to manufacture a working device, you need very little material with organic semiconductors, layer stacks with a total thickness of 500nm or less are fairly realistic for OLEDs and solar cells and you probably need even less for transistors. Compare that to the average thickness of your plastic / paper substrate and you can see that the layers would stay intact even for fairly small bending radii.
Usually organic semiconductors are made of polymers or large organic molecules, which have van der waals forces along their molecular chains, while inorganic semiconductors are made of ionic compounds, which have ionic interactions. When strain is exerted, organic molecules or ions are forced apart, long chains of organic s/c can still hold them together, while ionic interactions will disminish, so the inorganic s/c are broken.
Low temperature (usually lower than 200 oC) even room temperature processing technique could be conducted on flexible substrates, but the problem is flexible devices could not be bent, like Bu laju said, if you bend the flexible substrate, usually crystalline organic semiconductors on the substrate will also be damaged for their crystal islands could probably crack. Based on this, I think it is difficult for small molecules to realize real flexible electronics, more promising substitute is polymer, Bao zhenan recently published one science and one nature paper, one about self-healing polymer and one based on nanoconfiment, I think her work are more promising for industry applications.