Adafruit industries has open-source build-it-yourself kits of a very similar persistence-of-vision toy. Their version uses all through-hole components so it is easier to build, and it programs directly ...
I used a fairly generic circuit board I had lying around for this project. The circuit is really simple though, just an Atmel AVR microcontroller, 10 LED's, a programming header and a few resistors and capacitors. maybe a button if you want to be able to turn it off. The board I used is mostly surface mount components, but all the parts are available in thru-hole form so you could easily build this up on a proto-board.
I've attached below the Eagle CAD (http://www.cadsoft.de/) files for the circuit board, the C source code for the microcontroller, and the Gerber files of the circuit board to get it manufactured. Eagle CAD is free for small-size boards like this one.
Parts used: Atmel mega8L microcontroller - digikey ATMEGA8L-8AC-ND 6-pin .1" spacing pid header - digikey WM6806-ND 1206-size surface mount LED's in color of your choice - digikey 160-1406-1-ND, 160-1404-1-ND, 160-1402-1-ND 150 ohm resistor array: digikey EXB-V8V150JV 10uF 0805 size capacitor: digikey 587-1299-1-ND switch: digikey CKN4081CT-ND lithium-ion battery: from batteriesamerica.com (http://www.batteriesamerica.com/newpage8.htm)