during today's lesson I got a message from Adobe Flash Player saying Duolingo wants to access my camera/microphone. I clicked deny, then the record feature would not work. I had to skip and lose a heart??
You should be able to right click (cmd-click on a mac if right click is not set up) on the "record" button, then select "Configuration" (or settings or something).
In there, select the microphone tab, and click allow.
You should now be able to do the speaking tests. If you don't want to do them at all, click on your name up the top of DuoLingo, then Settings, and turn off the microphone / speaking tests.
How on earth did you think it would be able to record without using your microphone?!