I am not sure if the GH2 records in some Panasonic variant of AVCHD that may need to be transcoded by 3rd party software but first:
Make sure your copy of FCE has been updated to 4.0.1
Open Log and Transfer, and click on the preference gear in the upper center of the window, choose Preferences:
Then set the AVCHD audio to Plain Stereo:
Also, you should verify your camera storage card only has one kind of video on it, Log and Transfer has been known to act erratically if the storage device has SD and HD video or Video and Still Images on it.
If all this gets you nowhere then you may want to look at Clipwrap:
which will allow you to transcode the footage to AIC. I believe they have a free trial offer.
If Clipwrap does not work, you may have a corrupted storage card.