Here are a couple of the more common issues I've seen running Linux on the Bay Trail NUC:
The "no desktop" desktop: After installing Linux Mint you may see the Mint logo (i.e. the default background wallpaper) but no task bar or icons. This is caused by a "phantom" display. Here's how to get around it:
Right-click on the desktop and select Change Desktop Background, then clicking All Settings.
Scroll down to the Hardware section and open the Display settings. You'll see a phantom 1024X768 display set as default
Disable the phantom display and your HDMI connection will become the primary and everything should start working.
No audio through HDMI (many flavors of Ubuntu): This isn't specific to the NUC as it seems to be happening on lots of builds, but it's easy to get around:
Make sure your system is up to date using the Update Manager
Add the following groups to your account: audio, pulse, pulse-access, video, voice