I haven't looked in awhile, but ClamAV actually did very well in some third party virus tests. You shouldn't need a Linux specific product. Good AV systems detect viruses for multiple platforms irregardless of the host performing the scan. It's actually really hard to gauge AV effectiveness, so take those published tests with some healthy skepticism. ClamAV detects Linux and Windows viruses, but does not include an on access scanner. It's usually fired off manually or hooked via other programs (spam scanners).

There is however, a fuse FS ClamFS http://clamfs.sourceforge.net/ that provides on access scanning for Linux with ClamAV. I have no idea how much overhead that incurs, but it probably isn't a small amount. A lot of enterprise offerings provide on access scanning (McAfee, F-Prot, Kaspersky, and more).

I'd imagine there are Firefox and Chrome add ons as well.

You want maldet[1], it uses clamscan.

[1] - http://www.rfxn.com/projects/linux-malware-detect/

I sometimes get Windows virus attachments in emails, when clam doesn't identify them as a virus, usually avg can. I'd hope it can catch Linux viruses and trojans too.