POSSIBLE TEMPORARY FIX!
I'm having the same problem with my iPhone 6 plus, iOS 8.0.2. Worked fine for a couple of weeks and today suddenly stopped working. The on-screen wave moves as if it's listening, but does not type as I speak or even after I'm done speaking. I do have a few vehicles that I've been connecting to via Bluetooth, but no external keyboard as others have mentioned.
I re-started my phone by just holding the button on the right (top button on iPhone 5), then swiping across...no dice.
Then I followed the instructions someone else posted (thank you!) to do a "soft reset" by holding both the bottom button and the right-hand button and then swiping -- causing it to re-boot twice I guess. Anyway it worked after I did that.
I just sent a couple of texts and a couple of emails and so far so good.
Apple still needs to fix this of course, but this may get you working again at least temporarily.