Depending on which browser users are using, from the Tools menu, users should "Delete Browsing History..." in Internet Explorer 7 or "Clear Private Data" in Firefox. Once this is done, all functions of the Who's Online tool should work, including: The "Refresh Now" button, the "Auto-refresh" checkbox, and the "My Status:" update.
Side Notes: We have not yet determined exactly which parts of the history/data needs to be cleared, but I believe the browsing history, forms data, and passwords would not need to be deleted. Simply deleting cookies did not seem to work. Last time I checked the computer labs did not allow history, cache, cookies, etc. to be deleted.
Oh yeah! Users will have to have Java installed. The Check Browser available from the My Blackboard link should notify users of that.