Although Cortana for Android is not officially available from Google's Play Store, it has been in public beta since summer and many have downloaded the app from third party sources. Much like any beta software, it looks like the Cortana app might have a big problem, in that it could be disabling the mic on your Android device.
According to Mark Coppock of WinBeta, he found that his Nexus 5 was having issues with the microphone not working and almost decided to reset his device - until he found others with the same issue. With some experimentation, he found that Cortana was the culprit and that by disabling the app, it allowed his microphone to start functioning again. Naturally, re-enabling Cortana brought back the disabled mic issue.
If you are experiencing a similar issue on your Android device after installing the Cortana app, it might be worth uninstalling to see if it fixes your mic issue. It is unknown when an update for the app will arrive, but hopefully the update with fix will arrive soon.