The internal microphone on my Linux system didn't work. In order to enable it, I had to disable the driver for the internal analogue (classic) modem.

In order to check, if you might potentially be affected from the same issue, open a terminal, and run the following command:
* lsmod |grep snd_hda_codec_conexant

If the above command produces output, then your system has that particular Linux kernel module loaded, and might be blocking the internal microphone.

The solution is to prevent the system from loading the kernel module on startup. You can use the following commands:
* echo "blacklist snd_hda_codec_conexant" > /tmp/modem-blacklist.conf
* sudo cp /tmp/modem-blacklist.conf /etc/modprobe.d/

Now restart your computer and test if your microphone works.