The Sparkfun Geiger counter doesn't use an Arduino. But since they use the same processor it is easy to adapt. Using an Arduino makes it even easier. The chip for USB is already on the Arduino board, and the tranformer for audio out is not needed if a piezo beeper is used.
This is the link to the Sparkfun page : http://www.sparkfun.com/products/10742
They use the LND712 Geiger tube.
This is the page for the LND712 : http://www.lndinc.com/products/711/
The LND712 need 500V (recommended operating voltage).
Anyone knows a cheaper Geiger tube ? or an alternative way to detect radiation ?