Before you start, if you are having loading errors or random slowness, please check out "Is Duolingo down right now?" to see if we are experiencing server issues.
1. Check that you are using Google Chrome. Speaking exercises only work on Chrome.
2. Make sure the issue is not with the microphone itself. Confirm that it is on, plugged in, and that it works on other applications. You can also try refreshing the page or restarting your computer and browser to see if that helps.
3. Go to your settings page. Then make sure your microphone is toggled "on." Make sure you save your changes!
4. Assuming you are using Chrome, make sure Duolingo is on your list of websites "allowed" to use the microphone. If you denied it permission in the past, then you have to change it manually in your Chrome settings. More information on that here.
7. If it still doesn't work, consider submitting a bug report. You can disable the microphone (and prevent speaking questions from appearing) for the time being by setting it to "off" in your settings.