MEGA pushes the browser to its limits, and these limits vary. While it does work with all major current browsers, there are some weighty feature and performance differences:
Google Chrome: The leading browser, by far. It implements the proposed HTML5 FileSystem API, allowing for fancy features such as recursive folder uploads and efficient downloads. Caveats: Requires user permission to batch-write files after a few unattended completed downloads (for security reasons, and only once per session). Slightly anaemic text rendering.
Internet Explorer 9: Lacks all essential features required for MEGA: File I/O, Web Workers, ArrayBuffers, and binary cross-domain HTTP access. Nice text rendering, though.
Conclusion: If you are planning on using MEGA frequently, there is currently no alternative to using the most advanced browser currently in existence - Google Chrome.