my head canon is that in the Dune universe, their response to the Butlerian Jihad was to develop better and better mechanical computers; specifically, via miniaturization, down to the nanometer level. It doesn't quite work for everything (Holtzmann shields are entirely analog?) but it works well enough to map most objects to a viable analog controller made of nanometer-scale analog computers.
The ban on thinking machines in Dune has nothing to do with the mechanics by which those machines work.
For all we know (I'm ignoring Brian Herbert's fanfiction here) the predominant type of computing at the time was mechanical. In any case, it wouldn't have mattered.
