The calling thread cannot access this object because a different thread owns it

This is a common problem with people getting started. Whenever you update your UI elements from a thread other than the main thread, you need to use:

this.Dispatcher.Invoke(() =>{    ...// your code here.});

You can also use control.Dispatcher.CheckAccess() to check whether the current thread owns the control. If it does own it, your code looks as normal. Otherwise, use above pattern.

Another good use for Dispatcher.Invoke is for immediately updating the UI in a function that performs other tasks:

// Force WPF to render UI changes immediately with this magic line of code...Dispatcher.Invoke(new Action(() => { }), DispatcherPriority.ContextIdle);

I use this to update button text to "Processing..." and disable it while making WebClient requests.

To add my 2 cents, the exception can occur even if you call your code through System.Windows.Threading.Dispatcher.CurrentDispatcher.Invoke().
The point is that you have to call Invoke() of the Dispatcher of the control that you're trying to access, which in some cases may not be the same as System.Windows.Threading.Dispatcher.CurrentDispatcher. So instead you should use YourControl.Dispatcher.Invoke() to be safe. I was banging my head for a couple of hours before I realized this.

Update

For future readers, it looks like this has changed in the newer versions of .NET (4.0 and above). Now you no longer have to worry about the correct dispatcher when updating UI-backing properties in your VM. WPF engine will marshal cross-thread calls on the correct UI thread. See more details here. Thanks to @aaronburro for the info and link. You may also want to read our conversation below in comments.