What options are available to you on each monitor?
Which 7970 specifically?
A 7970 can certainly drive 3 displays without an issue, but there's likely limitations re: which outputs can be used together.
Gut feel on it would be:
DVI-DVI
HDMI-HDMI (or converted to DP)
DP-DP (or converted to HDMI)
Should drive 3 monitors. Now whether you have the ability to use one of each output will depend on the monitor.
I suspect the fact both DVI outputs are being used is limiting you somewhere along the line.
Try disconnecting one DVI output...... can you then use HDMI-HDMI?
According to this:
https://support.amd.com/en-us/kb-articles/Pages/AMDEyefinityFAQs.aspx
The 7900 series cards can drive 6 displays....... but not exactly sure how that's possible with only 4x display outputs.