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I have a GH2 and an OM-D. I only recently purchased the OM-D. The image quality from the OMD is significantly better, IMHO. The highlight roll-off is much better, a substantial difference. Also, the OMD noise in shadows is much better, IMHO.
I prefer the interface/menus on the GH2.
I prefer the multi-aspect sensor of the GH2 a lot, but the GH3 does not have a multi-aspect sensor unfortunately.
I like the grip/feel better of the GH2. The OM-D with RRS grip is pretty good though.
The GH2 is better for video, so everyone says. I only use them for stills.