You could run it, but I would
highly recommend against it. This isn't due to the raw number of frames per second you could get. Rather, the issue is frame time variance.
All AMD GPUs that use Crossfire bridges (including the 7970) suffer heavily from frame time variance issues with >4MP displays and Crossfire. Even though AMD has within the last year added some fixes for a small number of games, using Crossfire with older cards and >4MP displays is still fundamentally broken. This is the reason AMD replaced it with XDMA on the 290 series.
With a 4K (8MP) display and Crossfire 7970s, you can still
expect frame time variance of greater than 50% with the "fixed" drivers. For reference, less than 20% is considered acceptably smooth. In other words, games will look like a stuttering mess if you use 7970 Crossfire to drive a 4K display no matter how many frames per second FRAPS or some other utility shows you. Of all AMD cards, only the 290 series performance acceptably in Crossfire with >4MP displays in regards to frame time variance.
I have 3 7970s for a 5760x1200 resolution (7MP), and I can't stand to use them for Crossfire because it actually looks less smooth than a single card at a lower framerate. If I was in your situation I wouldn't get a 4K display unless you're happy with a single 7970 or you can replace your system with AMD 290 series card(s) or modern NVidia card(s).