I have a 5770 XFX XXX 1Gb RAM, and in my experience the answer is: no.
After taking a look at the official documentation the answer is still no.

The thing is that the radeon driver package, a.k.a. the open source driver for the ATI vga, offers some support for the oldest cards, for the new ones there are two main option:

• the fglrx package that basically contain proprietary driver but
  packed by the Ubuntu team ( tipically outdated package )




• the, so called, fglrx ccc AMD ATI proprietary closed driver ( that are the
  most updated drivers and probably the right ones for your card )

Downloading and building your own package from the latest driver available from AMD ATI is a pretty easy task http://wiki.cchtml.com/index.php/Main_Page and gives you good performance.