To use three monitors with AMD Eyefinity at least one of them has to use the display port (either mini or full). This table is in the wikipedia article for 5800 cards but it applies to the 6770 as well:
If you don't have a monitor with display port as an input you can get an active DisplayPort to DVI-D adapter. I have a 6950 with my main monitor using HDMI one side monitor using DVI-D and the other using this adaptor to DVI. It works very nice the only downside being that occasionally when I wake the computer from sleep the monitor on the adapter doesn't wake up. If I unplug and replug the adapter it works.
To use the onboard controller. Verify that both drivers are installed and up to date and Windows correctly lists the graphics devices in Device Manager. Test using only the onboard controller to ensure that video out is working from there.