I tried to compile my own binaries from source to avoid this, but I kept running into architecture errors and ended up throwing in the towel after a few hours of fiddling. Luckily, a ZSNES developer known as Nach has provided specialized precompiled binaries for a variety of architectures that worked a treat for me.
Just follow this link, scroll down a bit, and then download the binary that matches your architecture. I have an Athlon 64 X-2 4000+, so I selected the Athlon64-SSE3 binary.
Next, install libsdl-dev either through Synaptic or by typing sudo aptitude install libsdl-dev into a terminal.
Once downloaded and decompressed, you should find your binary inside, which you can run by double-clicking it or typing ./zsnes into a terminal. If you double-click and nothing happens, try running it through the terminal to spot any errors. I personally encountered this error:
Unable to poll /dev/input/event8. Make sure you have read permissions to it
repeated 9 times (1 for each /dev/input/event* 0-8).
I was able to get around this by running ZSNES as root (i.e., sudo ./zsnes), which is certainly not ideal, but I haven't found a way around it yet.
If you take this route, be aware that the default location for saved games will not be ~/.zsnes as it normally would be. Instead, it will be located in /root/.zsnes (since you're running as root). This in mind, you may have to copy your .srm files into this directory for them to be recognized. When I first got my copy to run, I couldn't get it to recognize any of my saved files, even though I tried changing the path for saved files under the preferences. However, once I copied the files into the /root/.zsnes directory, everything showed up just dandy.
Using the native 64-bit binary referenced above also had another unexpected (perhaps coincidental) positive effect of making my gamepad's d-pad work correctly. Using dfreer's zsnes32 binary worked well for me in most respects, but it just wouldn't recognize my Logitech Precision gamepad's d-pad. The buttons worked fine, but when it came time to assign the directions, it would just sit there dumbly, even though jscalibrator and cat /dev/input/js0 both showed plenty of action. This problem remained no matter how many kernel modules I loaded (usbhid, analog, etc.), until I tried Nach's binary. Now it works just fine. Go figure...
This information was written for Ubuntu 8.10 Intrepid Ibex, but it should be applicable to other distros as well.
