The short answer is no; there is no easy way to do this (as far as I know...I'd be VERY happy to learn otherwise!). This is the hardest part of web design to get right.
The long answer isn't all bad news, though. Although I don't do this for a living, I'd say as long as your site looks good in 800x600 and up, and works on IE 3 and Netscape 3 and up, you are way ahead of the game.
The best thing to do is have as many people on as many different setups as possible test the site before you go live. There are no doubt plenty of companies that will do this testing for you, but I imagine their rates are not cheap.
Someone else here at helponthe.net might know of some free online testing services, but such tests probably only help with browser compatibility issues.
In the end, regardless of how careful you are, you will be making some tradeoffs. You simply can't please all the browsers, all the time. There are too many flavors, including text-only browsers and wireless devices, and if you plan on using Java and/or Javascript or VBscript, things get even more complicated.
You might want to check here:
http://webdesign.about.com/cs/testing/
The about.com (and netscape.com) sites are a good jumping-off point for finding resources to help you.
Good luck!