On occasion, when trying to join voice in Discord, I hear what sounds like a
fast series of clicks (something like ~10 Hz). (I don't have a recording right
now, but the next time it happens, I will try to obtain one, if that would be
helpful).

When that happens, this is repeatedly printed in terminal:

ALSA lib pcm.c:8424:(snd_pcm_recover) underrun occurred

Closing and restarting Ripcord does not help. Killing and restarting pulseaudio does.

Other applications (I have tried at least mplayer) do not seem to be similarly
affected.

zeb@terabithia:~$ pulseaudio --version
pulseaudio 12.2
zeb@terabithia:~$ uname -a
Linux terabithia 4.19.24-1-MANJARO #1 SMP PREEMPT Wed Feb 20 22:59:23 UTC 2019 x86_64 GNU/Linux
zeb@terabithia:~$ cat /etc/lsb-release 
DISTRIB_ID=ManjaroLinux
DISTRIB_RELEASE=18.0.3
DISTRIB_CODENAME=Illyria
DISTRIB_DESCRIPTION="Manjaro Linux"
zeb@terabithia:~$ inxi -A
Audio:     Device-1: Advanced Micro Devices [AMD/ATI] SBx00 Azalia driver: snd_hda_intel 
           Device-2: NVIDIA driver: snd_hda_intel 
           Sound Server: ALSA v: k4.19.24-1-MANJARO 
zeb@terabithia:~$

There's not really anything I can do about this. Linux audio is impossible for a third party to help with due to the number of abstraction layers and various ways that people can have different setups on their own machine.

Ripcord already uses a fully threaded audio system -- the main thread and networking threads never block the audio thread. On Linux, it uses only the standard Qt mechanisms for doing audio and nothing else special. There's no way for me to troubleshoot this for you or to fix it, because Ripcord is already doing everything correctly, as far as I'm able to control it.