Sorry I posted this bug report here. The original post in the Ultrasonic forum (viewtopic.php?f=17&t=11303) has a link to submit bug reports that leads to this thread.
Anyway, I figured out what the issue is on my own. I assume more people are going to run into this with any of the Subsonic streaming apps on Android, so I'll post my solution here and will create a new thread as well.
The latest Android 4.4/Kit Kat update for the Samsung Note 3 and Galaxy S4 DISABLES the ability for "unofficial" apps (i.e. not pre-installed by your wireless carrier, Google, or Samsung) to write to the SD card.
Subsonic apps will install and work correctly if you leave them with their default settings to use your device's internal memory. However, if you specify your external MicroSD card (/storage/extSdCard) as your storage for any Subsonic Android apps, it will stop working with the following symptoms:
-The app will operate normally and will connect to your server and list the artists/albums as usual.
-As soon as you play a file, it will download and display album art, but the song won't play and either stops completely (the official app) or repeatedly count from 0:01 to 0:04 without playing anything (Ultrasonic).
Unfortunately, the only way around this is to root your device and restore the permissions for any apps to write to your MicroSD card with an app from the Google Play store called SDFix.
This is completely dumb. I'm not sure why Google and/or Samsung would do this, as it renders your MicroSD card useless for anything but what your carrier, Google, or Samsung deems appropriate.
Oh well. Hope this helps someone.
If you're looking for a method to root your device, I highly recommend XDA-developers.