Problem was that I had put myself on mute during the last phone call (a conference call). The phone seems to have kept itself on mute, even through 2 reboots. I had to make a phone call, put myself on mute and unmute myself, and then the mic worked.
Carlos' suggestion worked for me "Have you tried removing the Google account AND clearing the data for the Contacts Storage app (Settings, Application Manager, All) ?" - clearing the data was the bit I'd not tried before.