What is the Model and manufacturer of the 5770 card and what kind of display outputs does it have? According to AMD's web site here ( http://www.amd.com/us/products/desktop/graphics/ati-radeon-hd-5000/hd-5770/Pages/ati-radeon-hd-5770-overview.aspx#2
) the 5770 can do three displays.
However, as greghome said, you need three digital outputs such as DVI, HDMI and displayport. You will need to check the specs of your specific card to be sure that your card has (1) enough digital outputs and (2) actually supports Eyefinity (AKA multiple displays).
Your old displays likely use VGA connectors which are Analog. That means that you would need an active digital to analog convertor to drive one of your VGA displays (such as this http://www.newegg.com/Product/Product.aspx?Item=N82E16812270478).
You may also need to use a DVI to VGA and/or a Displayport to VGA adapter as well for one display. In my mind that's just way to many adapters and way too much money to spend just to support a bunch of old monitors.