Well my take is that no AV software is absolutely necessary on any platform. I liken it to driving a car with a big spike in the middle of the steering wheel. Stops you being careless.
I use windows with no AV software, but then I'm extremely conservative about what I choose to install.
Unfortunately not the best solution. Especially on Windows.
Worms, running on an infected system that can see your computer, will try to infect you. Yes you could rely on keeping the operating system and applications patched to the latest levels and having a firewall installed, but that doesn't guarantee anything. I heard comment recently that the intrusion detection on McAfee VirusScan* will stop around 80% of the viruses that exploit the bugs, so just running that AV product on an unpatched version of windows will give a great level of protection; it also means that if a virus comes out that exploits a new loophole, you're likely to be protected BEFORE the software author releases the patch.
Similarly there are a lot of viruses that are embedded in documents. Just opening an infected document (e.g. word or excel) can be sufficient to get the infection, which will then transfer to every document you subsequently open - and if you pass those onto somebody else then you're passing on the infection.
Even if you're running linux, so not affected by the embedded dodgy code, you could be receiving the virus from user A and passing it onto user C without realising.
Recently I've seen a lot of viruses going around which get through the firewall as they use MSN or Yahoo messenger to spread (other messengers may also be vulnerable). Any hole you've opened in your firewall is a hole that can be exploited. Users of P2P networks (music sharing, bit torrent, etc etc etc) are particularly vulnerable to this.
Even basic technology is being exploited again now. You know the "autorun.inf" file that enables a computer to automatically run an application on a CD/USB key/etc as soon as it's inserted? There's quite a lot of autorun worms that are going round at the moment, and the first thing they do is copy the virus executable and autorun.inf file to every single drive the computer can see. One infected USB key can immediately infect one computer and all its mapped network drives, and as soon as any other computer that maps to the same network drive opens that drive, the worm moves onto all new network drives it sees (this includes UNC shares).
I don't think I'd ever run a computer without AV installed. Even if I ran Linux I'd have something running, whether in the hope of preventing something such as Greenbank's comment above (and he's not the only person to think of writing dodgy software), or more importantly to prevent me passing viruses on to friends/relatives/etc that use windows (and the main reason most people I know use windows is that it's easy to use and they have to use it in the corporate environment, so stick with what they are used to instead of learning a new operating system).
* I use the corporate edition. The home user version sucks
