Hi GSlayerBrian,
Yes, that's how I have done this before. When doing it this way, you can disconnect one of the power sources as long as the other is connected. There will be a voltage drop across the diodes, and likely also a bit over the voltage regulator, so the PSU and battery will both need be be a little bit higher voltage than if they where connected directly.
For the Galileo Gen 2 you can probably omit the voltage regulator, as long as you keep the PSU and battery within the input specs of the Galileo, but Gen 1 will need a 5V regulator as indicated in your drawing.
Looking forward to hear how it goes.
/Thomas