I'm in the process of running open-source RadeonSI Gallium3D tests on the new MSI Radeon R7 370 4G graphics card. That initial Linux review of the AMD Radeon R7 370 with the open-source driver will be published later this week (still waiting on an updated Catalyst driver for those proprietary driver tests). However, as an excerpt of the Gallium3D testing, here are results from some AMD graphics cards with Counter-Strike: Global Offensive and Team Fortress 2.

In the testing so far has been a Radeon HD 6870, HD 6950, HD 7850, R9 290, and R7 370. There will be more cards in the R7 370 Linux review along with some fresh open-source NVIDIA benchmark results. Tests were done on Linux 4.1.1 and Mesa 10.7-devel atop Ubuntu 15.04. Linux 4.2 Git couldn't be tested (plus the R9 285 with AMDGPU) since this particular test system is still plagued by the Linux 4.2 kernel panics.

With the higher-end graphics cards, the current open-source AMD Linux stack does deliver a playable CS:GO experience on Ubuntu at 1080p.

With Team Fortress 2 it's less demanding and becomes CPU bottlenecked.
As the last benchmark teaser for now, Unigine:

The benchmark results are in this OpenBenchmarking.org result file. Stay tuned for the full graphics card round-up in the days ahead on Phoronix.