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There really is no short cut to cross-browser testing - because an emulator could not capture all the small peculiarities of each browser. The only way to know for sure is to do what you've been doing.
However, if you validate your code before you test on the various browsers, you have a head start. Then, as you test pages over time, you will build up a knowledge of all the little things that can trip up this browser or that. Soon, you're cross-browser testing becomes much quicker.
HTML Validator - W3C [validator.w3.org]
CSS Validator - W3C [jigsaw.w3.org]
To validate CSS on a file that is already online I usually use:
[jigsaw.w3.org...]
You can get to these pages in a few clicks from the links that Tedster posted, but I have bookmarked these alternative direct URLs on my system.
It will let you test different levels of HTML compatibility, different screen sizes and even let you see how a search engine will view the tags on your site.
Chris.