After installing I had this issue with Manjaro. I had the same distro installed with a 660ti and it worked. Tried reinstalling it and it did not work, it gets to the log in screen but does not properly display things. If you find a solution please tell me!
I had the same issue with an HD7870XT (GCN 1.0). The graphical installation would work, but after a reboot I'd get a black screen. I was however able to get it running on arch linux with fglrx (the outdated proprietary AMD driver). The open source drivers wouldn't work on arch for me either. However these old proprietary drivers are suboptimal. I had poor html5 performance in firefox and had to use an old xorg version. I decided to rather run with my (much less performant) integrated r7 graphics in the A10 7890k.
Since your card has the GCN gen 2 architecture there is experimental support for the amdgpu driver that would have to be manually enabled in the kernel. But that is seems to be a rather complicated procedure. I'd also believe that you won't have much luck, if not even the older (but still maintained) radeonsi driver works... :/ let's see what @Eden has to say.
If it's a display issue, I'm thinking it's most likely a driver, kernel, or hardware issue. Have you tried installing Manjaro using the Non-free drivers? Using one of the official versions featuring either Plasma or Xfce may work. If it does, the issue is almost certainly that it need the proprietary drivers. Not sure why that would be, though...(Non-free drivers are proprietary (should come with Catalyst Control Center, if not Crimson))
AMD has largely only been working on the 480/Fury side of things, the 290/390 cards often get forgotten and BIG regressions and bugs crop up within MESA often.
However in saying that Blackscreens can occur on NVIDIA cards, for example my Linux install atm has a blackscreen with my two 980Ti cards because I enabled multi-gpu mode which apparently x.org doesn't like (I was trying to get SLI working which apparently is a major undertaking to enable correctly for Linux)
I have an R9 390, and I installed several different distros with it before finally landing on openSUSE, and I don't recall having any trouble with it. I believe Fedora and Manjaro were among the ones I tried, and had working.
With my 390X, I sometimes have to add radeon.dpm=0 to my boot script in order to get passed the black screen thing. It's been resolved with distros with newer drivers/software though. Out of all the distros I've tried, Antergos has worked the best for me with my card.