As far as I know there is no GPU that will support three displays without at least one of those displays being connected through DisplayPort or mini DisplayPort. Now, you can buy an adapter for DVI or HDMI to Display Port. Those have to be active (not passive as most cables are) and will run between about $25-$30. I'm not certain whether your given card can support triple monitors, you will have to check the documentation, but looking at the general details on AMD's site, that isn't going to happen with that card.
So, you have two options. There is an option you can try, using a
Display Link kit. There are a number of other companies that make similar devices (I just happen to have received this one a year and a half ago). You basically connect the monitor to the device, and the device connects to your PC through USB. It's very convenient for what it does, and in general costs less than buying most new GPUs.
Or, if you want a new GPU you will need at least a Radeon HD 5000 series. Don't forget, however, that you will still need either a monitor with native Display Port capability or an active DP adapter cable (
something like this one).
Hope that helps.