Hi,
You basically have a few choices:
1. Pick a different distribution of Linux (including older LTS Ubuntu) that provides fglrx directly.
2. Install fglrx directly from AMD. (This is what I do on my Linux Ubuntu box - this lets me run the latest Catalyst driver that matche Windows, more or less.)
3. Run with an unsupported (by Laminar Research) open source driver and hope for the best.
If you want to try option three, I suggest reading the Linux forum at X-Plane.org - other Linux users have gotten this to work. But this is an unsupported configuration; if X-Plane won't run with a particular open source driver (or update to a driver) or stops working due to an update to the sim, we are not going to be able to fix it for you. (We do try to collaborate with Linux developers who are working on the drivers, but it's not the same as our guarantee that it -will- work that we can provide, for example, with the latest Windows proprietary drivers.)
Unfortunately, it is beyond the scope of X-Plane tech support to help Linux users configure Linux. Basically if you have a working recent fglrx running in direct rendering mode, we expect X-Plane to work and can look at issues.