Unigine Heaven is the most demanding OpenGL Linux benchmark at this point. At 1920 x 1080, the Sapphire Radeon HD 6770 in Unigine Heaven is just under 20 FPS while the Radeon HD 5830 is at 22 FPS and the Radeon HD 6870 is up at 27 FPS.
The test results from the Sapphire Radeon HD 6770 are not to any surprise considering the Juniper core at its heart. Regardless of being a rebranded previous-generation ASIC, the performance of the Radeon HD 6770 is great, especially if your purchasing decision comes down to choosing between the HD 6570 or HD 6770. Going with the Sapphire Radeon HD 6770 will give you the slight (50MHz) stock overclock and a quiet but effective cooling solution.
Sapphire is also one of the most reputable AMD AIB partners and through their support of Phoronix via hardware samples are indirectly supporting the Linux hardware community. At the moment, this graphics card is selling at a point just above $120 USD, which is right around where the Radeon HD 5770 cards continue to sell. One of the benefits of the AMD HD 6770 not being a new GPU design based on the "Northern Islands" is that the open-source support is more mature and better off right now for the Radeon HD 5000 "Evergreen" generation of hardware, for those interested in using the alternative Linux driver.
For those evaluating a graphics card upgrade and wishing to see how your system's GPU performance compares or to that of other systems, the results from these eight AMD Radeon HD graphics cards are available via OpenBenchmarking.org with the 1105307-GR-RADEONHD605 result file in conjunction with the Phoronix Test Suite software.