When you access Google from the web, all it can check is your IP address, your cookies for all Google-owned domains (i.e. YouTube & Picasa count), and your Flash Local Storage (clear it at
Adobe - Flash Player : Settings Manager - Global Privacy Settings Panel). You don't need a new PC or router or Internet connection -- just to dump all your cookies & your Flash Local Storage, then go through a proxy.
If you run AdWords Editor, on the other hand, it collects all sorts of info about your PC, including the serial number on your hard disk, so to clean that up you need either a new PC or to use a VM. But as long as you only use the web site, IP+cookies+Flash is all you need to change.
Of course, that's just technical tracking mechanisms. Where most people screw up is not in the things Google tracks, but the info we flat out give them. Name, address, billing info are the obvious ones, but they also ban domains like crazy. If you're linking through the same Prosper202 domain, or directing traffic to any of the same domains, they'll catch that instantly.
When I was working with Google Base, once they banned an account I couldn't even link to a domain that contained
images from one of my banned domains, or which contained links to my Prosper install, unless I changed those domain names. Otherwise the new domains & accounts would be banned instantly. I wouldn't be surprised if they do something similar w/AdWords.