The author of Jukebox 2012 has updated it, and it now works the way I wanted it to.
With the startup options set to "Random Artist Sort" and "Random Start Position", the screen will display a random set of songs from your music library on your phone/tablet. Using the 'next' arrows, another random group will appear.
Until this update, you were basically presented with a long series of screens in alphabetical order.
For some reason, the app does not find most of my music. Even when I just copied 36 tracks to the MUSIC directory of the phone itself (not to the external SD card) and pointed the app at the MUSIC folder, it only showed 13 out of the 36 tracks. One album that I set it to only had one track display. Many albums say "no music found" when I point it to them.
I have emailed the author, asking if he's aware of any issues like these. Maybe it's a bug that can be fixed. If so, this app would be a fine jukebox app for Android tablets and phones.