To make your design cross browser compatible, make sure you design it using the W3C standards. As long as you comply with those standards, your design should work in all browsers.
Essentially, the most important thing for a developer to do in terms of cross-platform support is test the site themselves. My development workstation has every browser you can think of installed, including some of the most outdated (original Netscape... haha) and obscure pieces of software out there. When my company finishes a site for a client, there is an entire day devoted to manually checking different browser compatibility, along with frequent checks during development.
It's often overlooked, but there is no substitute for just going through your site yourself on as many browsers as you can.