1. Copy the game folder "Battlefield 3" from Origin <install path>\Origin Games\Battlefield 3
2. On the machine that doesn't have BF3 installed start the download and pause it immediately (when it gets to 0.01%), this will give origin a chance to create the folder on your computer.
3. Replace the "Battlefield 3" folder with the one you copied.
4. Resume installation, if everything is right, it should jump to 99% and download a couple of things and you're done!
Feel free to post your own tips if you didn't do it this way.