I'm using a wire harness, and matching things up. It seems like the stereo will power up and then die, I tried a second unit to see and it was the same thing. Any ideas? Seems like it doesn't have enough voltage (multimeter is reading 12.5 though)

 

have u checked thru the wiring diagram?

Foxed.ca - Mazda RX-7 Manuals

 

Is there an inline fuse on the HU or somewhere on the wiring? Try to replace it if there is.
Like Michael mentioned above, re-check the wiring.
Keep us posted on the progress.