The main difference between the Dispatcher and other threading methods is that the Dispatcher is not actually multi-threaded. The Dispatcher governs the controls, which need a single thread to function properly; the BeginInvoke method of the Dispatcher queues events for later execution (depending on priority etc.), but still on the same thread.
BackgroundWorker on the other hand actually executes the code at the same time it is invoked, in a separate thread. It also is easier to use than true threads because it automatically synchronizes (at least I think I remember this correctly) with the main thread of an application, the one responsible for the controls and message queue (the Dispatcher thread in the case of WPF and Silverlight), so there’s no need to use Dispatcher.Invoke (or Control.Invoke in WinForms) when updating controls from the background thread, although that may not be always recommended.
As Reed said, Task Parallel Library is a great alternative option.
Edit: further observations.
As I said above, the Dispatcher isn’t really multithreaded; it only gives the illusion of it, because it does run delegates you pass to it at another time. I’d use the Dispatcher only when the code really only deals with the View aspect of an application – i.e. controls, pages, windows, and all that. And of course, its main use is actually triggering actions from other threads to update controls correctly or at the right time (for example, setting focus only after some control has rendered/laid-out itself completely is most easily accomplished using the Dispatcher, because in WPF rendering isn’t exactly deterministic).
BackgroundWorker can make multithreaded code a lot simpler than it normally is; it’s a simple concept to grasp, and most of all (if it makes sense) you can derive custom workers from it, which can be specialized classes that perform a single task asynchronously, with properties that can function as parameters, progress notification and cancellation etc. I always found BackgroundWorker a huge help (except when I had to derive from it to keep the Culture of the original thread to maintain the localization properly :P)
The most powerful, but also difficult path is to use the lowest level available, System.Threading.Thread; however it’s so easy to get things wrong that it’s not really recommended. Multithreaded programming is hard, that’s a given. However, there’s plenty of good information on it if you want to understand all the aspects: this excellent article by our good fellow Jon Skeet immediately jumps to mind (the last page of the article also has a good number of very interesting links).
In .Net 4.0 we have a different option, Task Parallel Library. I haven’t worked with it much yet but from what I’ve seen it’s impressive (and PLINQ is simply great). If you have the curiosity and resources to learn it, that’s what I’d recommend (it shouldn’t take that much to learn after all).