As long as you stick to the APIs implemented in each browser, your code can be exactly identical. Realistically, you'll probably need to add some polyfills and edge case handling for IE, but it's much less than developing for different full-on platforms.

With the major exception that all evergreen browsers (most browsers in wide use today) quickly officially support new web standards, but Windows, OS X, and Linux are unlikely to ever officially support each others' native toolkits.