Which application code are you trying to use? The project in the example/ directory? I just checked the example in both the master branch and v2 branch on github, and both work on my 32a4u board. I'm compiling with the Atmel binary toolchain from
http://www.atmel.com/tools/atmelavrtoolchainforlinux.aspx.
The v2 branch has optimizations and an improved API. We're using it in the new CEE firmware 1.2. Since it's stable, I will be merging it to master shortly, so try that branch and see if it works better.
The build system is a bit of a hack (especially in the old branch) as it tries to support older compilers that don't have native support for the USB-enabled parts (it compiles as 32a4 but provides its own 32a4u header). In the old master branch there was another mention of the part name in the USB-XMEGA/makefile that shouldn't have been there. The v2 branch supports changing the part name more easily, and comes with headers for all USB-enabled parts.
I've used jumper wires with low speed USB before. Full speed would probably be fine too, but I don't know.
Hope that helps, and thanks for reminding me to get more documentation up so you and others can use the USB stack more easily.
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Kevin Mehall
Nonolith Labs