"cross-browser" compatibility means compatibility with all of the major web browser software. Today any web site that uses the normal standards of modern web design, like HTML 4.x, XHTML, Javascript, CSS, and Flash, is going to work just fine with the major web browsers (Internet Explorer, Mozilla, Safari, and Opera). Not all of your style sheet settings are likely to work perfectly with all browsers unless you actually test your pages with multiple browsers, which is recommended, but it is rare for a page to be unusable with any major browser unless it relies extraordinarily heavily on Javascript and no actual testing has been done. Just follow these recommendations:

Test your site with Microsoft Internet Explorer and Mozilla, at a minimum.
Include plain-text links to your content and text alternatives (alt attributes) for your images that contain text or are crucial to understand the page. Otherwise browsers for the blind will not work. Equally important, search engines will not index your page properly.
Don't build your whole site in Flash. While most people have Flash, browsers for the blind, search engines, handheld devices and users who do not like Flash will completely miss your site. You should use Flash for accent, pizzazz, and entertainment, not as your entire design.