I have a degenerative disc, it was diagnosed 20 years ago. At the time I was advised not to run or do any kind of exercise which would put pressure on it so I took up swimming and that was pretty much the only exercise I did for years. I decided that I would try running last year when my back had been pretty good for a while and I can honestly say it has helped. I saw a physio before I started increasing distance and he was fine for me to run.
I was under the impression that the advice with back problems these days was to keep mobile so I am surprised you have been told not to run? See a good physio and see what their advice is and definitely ask the surgeon what his reasons are for the advice he (almost definitely a he!) gave.