I have a Radeon R9-270 and I am running Fedora 23 with the default OpenSource
drivers supplied with the repository.

Yesterday I did an Update from Kernel 4.3.3-300 to 4.4.3-300 and I started
seeing some issues. When I do nothing for about 12 seconds the screens (dual
head) turn blank. Moving the mouse makes the screen go back on again. Sometimes
the login-screen of the gnome-shell or chrome are frozen but the system as a
whole seems fine.

During normal operations /sys/kernel/debug/dri/0/radeon_pm_info is the
following:
power level 0    sclk: 30000 mclk: 120000 vddc: 950 vddci: 1000 pcie gen: 2

When the screen is still for a bit (exactly when it turns blank) the output is
power level 0    sclk: 30000 mclk: 15000 vddc: 900 vddci: 850 pcie gen: 2

So apparently, when the video card is clocked down, the output is deactivated.

My current "workaround" is running videos on the side - not an ideal solution.