I now have the Simon game working without an Arduino board! I wanted to be able to make several running projects without having to buy a board for each one, so I followed this guide: http://arduino.cc/en/Main/Standalone
In the photo above, the large circuit board is performing all the functions that the Arduino used to, including providing regulated power, a clock, a status LED, and connections for the I/O pins of the ATMega chip. The ATMega standalone chip with the Arduino bootloader preloaded can be purchased from sites such as sparkfun, or you can burn your own bootloader.
I can now program the chip in the Arduino then use it in an independent project. Now I just need to clear off these breadboards for my next project by soldiering it all on a PCB!