How to shield multiple flat (ribbon) cables properly?
Ideally, put a Faraday cage (a grounded metal box) around everything from the back of the computer to wherever they are going and solidly connect it to the computer earth. 

Ribbon cables are not really designed for use in the open. 

There are special twisted pair ribbon cables, but that only works if the pinout is right, and it would not generally be helpful if it was not designed in. 

As you've found, common mode chokes (the ferrites) help a bit. 

The above are just general suggestions for cables carrying logic-level signals. It might be a bit different if the cables were carrying floating signals or had other unusual characteristics. 

Differential signalling is good because the return path for the current can be twisted with the other wire so the net field is very low.