Thank you for your response. Your sample worked perfectly, so I started trying to determine the difference between our projects. I changed a few things in the web.config, but that didn't work. The projects are nearly identical at this point, but mine still behaved in the manner I described.
Finally, I tried using the Telerik.Web.UI.dll that you sent with your sample, and that did the trick. I compared the version numbers; mine was version 2008.2.826.35, and yours was version 2009.2.713.35. Obviously, mine is a whole year older than the one you provided. Has the product changed significantly since then with regard to the LoadingPanel? I'm working with an existing system, and paying to upgrade to the latest version of the product is not really feasible.
Any guidance you can offer is much appreciated. Thanks!