I've tested the OMD vs. the GH3 4 times, just bought the gh3.
The OMD focuses as fast and well as any camera I've used except my Canon 1dx and it's damn close to the 1dx.
The only problem with the OMD is it doesn't track focus as well. It hunts some then locks.
For catching that one image, it will do it fine, but for a succession of images where the subject is moving quick, it will miss some.
Olympus knows this and I believe the next OMD will have dual focus aids and they say a Sony sensor.
The Olympus does have amazing image stabilization.
The GH3 doesn't snap to focus as quickly as the OMD, but will track with reliability, in still and video.
In video it autofocuses amazingly well.
The best thing to do is to test it. You can rent them from borrow lens or one of those online companies fairly inexpensively.
Now, I really like the gh3, though the omd has a better still image from the same sensor and I might buy the OMD for stills to go with the gh3.