From Sofia to the West of Turkey should be relatively easy. After that, travel through Iran, Afghanistan and Pakistan will get hairy.
10s of thousands of religious pilgrims travel by bus between Pakistan and Iran every year. You can just avoid Afghanistan.
There were some bad roads in the Balkans as well.
You'd be surprised at how cheap, relatively safe and reliable bus services are in those regions.
Source: me and my wife traveled extensively by public transport in, well, at least Pakistan. The other countries are indeed sort-of hairy, but mostly for job-clearance-related reasons.