As akjoshi and Julio say this is about dispatching an Action to update the GUI on the same thread as the GUI item but from the method that is handling the background data. You can see this code in specific form in akjoshi's answer above. This is a general version.
myTextBlock.Dispatcher.BeginInvoke(System.Windows.Threading.DispatcherPriority.Normal,
new Action(delegate()
{
myTextBlock.Text = Convert.ToString(myDataObject.getMeData());
}));
The critical part is to call the dispatcher of your UI object - that ensures you have the correct thread.
From personal experience it seems much easier to create and use the Action inline like this. Declaring it at class level gave me lots of problems with static/non-static contexts.