Following my post here, I'm turning this into a tutorial:
I cannot say that this tutorial will work on other Mac hardware, or with other Ubuntu Mate versions, OR with other Ubuntu versions in general. If you try it and it works with other versions/hardware, feel free to chime in!
To single boot Ubuntu Mate Linux 16.04 Beta 2 on a 2007 MacBook Pro(3,1):
To create the USB media, simply dd the iso to the USB stick, it will just work:
dd bs=4M if=path.to.iso of=/dev/sd_ status=progress && sync
When the MBP boots, hold the "Alt" key until boot options come up, choose "EFI BOOT"
When grub comes up, edit the grub menu and add "nomodeset" to the grub line, then boot. Otherwise I got a black screen.
When installing Ubuntu Mate 16.04 Beta 2, the default partitioning options did NOT work for me (the system did not find any bootable devices). I did one giant partition which had / and /boot/ and /home etc etc, then 1 swap partition. This worked.
I will edit this post with additional tips/tricks as I experiment with it!