So Ryan has been playing many games lately, as a comparison between the latest GPUs from AMD and NVIDIA. While Civilization: Beyond Earth is not the most demanding game in existence on GPUs, it is not trivial either. While not the most complex, from a video card's perspective, it is a contender for most demanding game on your main processor (CPU). It also has some of the most thought-out Mantle support of any title using the API, when using the AMD Catalyst 14.9.2 Beta driver.
And now you can!
The Catalyst 14.9.2 Beta drivers support just about anything using the GCN architecture, from APUs (starting with Kaveri) to discrete GPUs (starting with the HD 7000 and HD 7000M series). Beyond enabling Mantle support in Civilization, it also fixes some issues with Metro, Shadow of Mordor, Total War: Rome 2, Watch_Dogs, and other games.
Also, both AMD and Firaxis are aware of a bug in Civilization: Beyond Earth where the mouse cursor does not click exactly where it is supposed to, if the user enables font scaling in Windows. They are working on it, but suggest setting it to the default (100%) if users experience this issue. This could be problematic for customers with high-DPI screens, but could keep you playing until an official patch is released.
You can get 14.9.2 Beta for Windows 7 and Windows 8.1 at AMD's website.