A former co-worker of mine introduced me to a site that did this that I liked quite a bit not so long ago while we worked on a project that yet hasn't been uncovered for the public eye. I'll ask him for the link (clumsy me didn't label the bookmark properly) later today.
Edit: http://browsershots.org/ (found it anyway ^_^)
There's plenty of sites like this around, some are free, some aren't. Usually the ones that aren't free gives you more options (resolution for example).
In my life of experience I'd say that nothing beats a good friendship network of computer users on various *nix distros, Mac OSX users and of course Windows users. Depending on which kind you are, download as many browser versions you can (including stand-alone copies of older versions if you suspect people of using it). I'm in Windows, and nowadays there are about 5 billion browser versions out there. Luckily, most of them run on the same engine (or modified copies of them) and generally handle things the same. I run anything I code (client-side that is) through a stress test (stretching and bending the site, flooding AND drying out the content areas, resizing the font, etc.) with the following browsers: Opera (the latest copy, Opera users tend to be quick with the updates), Mozilla Firefox (Mozilla works just as well though), Microsoft Internet Explorer (including stand-alone copies of older versions).
When my codes pass through these tests, I usually just get a few random errors reported through my testers on other OS's which I can fix quickly.