iPad 3 here. iOS 6.0.1. 32 MB WiFi + 4G. Using wifi now. Similar problem. Dictation works intermittently.
Using the Apple Notepad, dictate a sentence or two. Then go back to reposition the cursor and dictate another sentence or two. The dictation system hangs with the flashing circles, never to return. Dictation lost. Wait 5 or 10 minutes, dictating works again.
Scenario is repeatable about 95% of the time. This has been happening for at least six months. Reported to Apple feedback today.
So I just had the same problem and it was driving me crazy, and just when I was submitting my feedback to Apple I realized that the key clicks also stopped whenever my dictation stopped working, and also before dictation stops working I would never hear the final DING sound when I stop the dictation. So I tried to TURN OFF KEYBOARD CLICKS in the settings>sound. This worked for me! Good luck!