Diehard IE adherents who want Greasemonkey functionality can't do better than Trixie. Just download and install it, and it will show up in the IE Tools menu. You can either grab scripts made for Trixie or go to the main Greasemonkey script repository, Userscripts.org, and try those out. Many of the authors test for Trixie compatibility as well.
If you're still using IE 7, you might want to try out IE7Pro, which offers similar support for user created scripts. Though some Greasemonkey scripts will work on IE7Pro, compatibility isn't guaranteed. Certain aspects of IE7Pro will work in IE 8, but it's not completely implemented yet.
Until very recently, Google Chrome's support for user scripts was a for developers only thing, and support was scarce. Now, however, it's pretty easy to enable: Just head over to the Google Chrome Extensions page and download the Google Chrome beta, and you'll be ready to go. Google-supported items are plentiful on the company's own Extensions page, but many of the Greasemonkey scripts from Userscripts.org will work for Chrome, too.
Apple hasn't included any official support for user created scripts for Safari on Windows or the Mac, but Mac Safari can use a combination of SIMBL and GreaseKit to get standard Greasemonkey scripts to work. Mac Safari users can also take advantage of the plug-ins and AppleScripts at Pimp My Safari.
Unfortunately, no equivalent SIMBL/GreaseKit combination exists for Windows Safari users, so you're probably better off switching to Firefox (if you want Greasemonkey-specific support) or Chrome (if you want to use another WebKit browser with promising extensions).