This has been suggested to fix dictation not working.
Settings>General>Reset>Reset All Settings. You will have to enter all of your app preferences and device settings again.Then go to Settings>General>Keyboard>Dictation>Off. Then turn it on again and wait until it connects to the Apple Server again. Then try again.
Before that Herculean (and presumably time-consuming and annoying) method, try this:
I was having problems with dictation for at least a week -- the 3 dots would just vibrate a second, and then just disappear. It wasn't an internet connection issue at all; this happened whether I was on wifi, or on cellular with 5 bars and 3G. I finally found a (bizarre) solution on another site -- use Siri to dicate a text (e.g., activate Siri; tell it to send a text to ________; she'll ask you for the content, and you say what you want the text to say), then tell her to send it.
Voila, now if you go to use dictation, it works again. (At least, this worked for me -- I tried dictating an email and it didn't work; used Siri to send a text; then went back and tried dictation again and it was working perfectly.)