I just put the finishing touches on my Raspberry Pi 3 emulation machine running RetroArch. I was not a huge fan of RetroPie due to the reliance on Emulation Station - more moving parts meant that there were more things that could potentially break. I just wanted something that would run raw RetroArch, no frills.
This tutorial is mostly recreated from memory and was most recently tested with a Raspberry Pi 3 running Raspbian Stretch and RetroArch 1.7.7. If there is a mistake or a broken link, PLEASE message me and I will fix it.
I used Raspbian Stretch Lite from this page. Write the image to your SD card using something like Win32 Disk Imager, or if you're using OSX/Linux follow a tutorial on how to write the image using dd.