SteveG47 over at Microsoft Answers provided a solution.

  1. Go to Control Panel\Hardware and Sound\Power Options\System Settings
  2. Click on Change settings that are currently unavailable
  3. Uncheck Turn on fast startup (recommended)
  4. Restart your computer

Below is how the Recording tab in the Sound control panel looks when the Surface Pro 4 Microphone is working properly.

Note the level indicator on the right. When the mic is not working, that indicator is either absent or unresponsive.