It took me hours to resolve this, but I finally found a solution, so I'll quickly summarise my findings for the benefit of anyone who uncovers this thread through google like I did.
The ATI Radeon HD 5470 graphics card does work with the open source driver, but will likely make your laptop run 25C hotter than it does in Windows (try programs like CPUID hardware monitor in windows / Psensor in Ubuntu to check this yourself if you like). For me, this took the graphics card up to 80C+ when idle and made the laptop too hot to touch.
To bring the temperature down you need to use a different graphics driver. Here's where the fun begins...
There are two main desktop environments for ubuntu: UNITY and KDE. If you download "standard" ubuntu from http://www.ubuntu.com/download/desktop then you'll get UNITY. If you download "Kubuntu" from http://www.kubuntu.org/getkubuntu you'll get KDE. It is also apparently possible to install KDE after booting Gnome and vice versa...but I haven't tried it, and that's another thread.
Both Unity and KDE are great. If I had the choice, I don't know which I'd choose but the important thing is:
UNITY is incompatible with the proprietary fglrx driver for the ATI Radeon HD 5470 graphics card.
You need this driver. I'm unaware of any other driver that works and keeps the heat at a manageable level ...which basically means you can't use UNITY, and you'll have to use KDE.
SO. To business...
EDIT: this definitely works on 12.04 LTS, but may work on other versions too.
1. Install Kubuntu.
2. Remove any existing fglrx drivers and their settings:
3. Install the required driver:
sudo apt-get remove --purge fglrx*
4. Configure the driver
sudo apt-get install fglrx fglrx-amdcccle
sudo aticonfig --initial
6. To get full hardware acceleration:
7. Enjoy Ubuntu!
sudo apt-get install xvba-va-driver libva-glx1 libva-egl1 vainfo
Thanks to QIII for providing a version of this solution on another thread.