This morning I decided I was going to get my laptop and try hooking my Galileo up to it and see if it was just an issue with my Windows 8 system (though it was working before). When I plug in my power supply into the Galileo it doesn't power up, it was working yesterday. When I plug it into USB it will power-up. I was excited about the Galileo but from what I am reading and my experiences its not good. I exercise safety when working with it too, I always wear a ESD wrist strap and never set it down on anything but an ESD mat. I have checked my power supply and it reads 12v, when I check the voltage on the +ve/-ve posts when connected to the board it reads between 20mV-200mV.
The box says what will you make, so far all I have made was 3 posts to an Intel forum.
I have read your other thread: Technical Questions and we have received your warranty ticket as well.
Regarding your question about the LM317, I think it has an output current up to 1.5A. The power supply unit I use for the Galileo Gen2 has also an output current of 1.5A so I think you can use the LM317 to power the board.
About the issue with your board, as I said, we received your warranty ticket so you will receive support in it.