I need some resources to get started on using neural networks for time series forecasting. I am wary of implementing some paper and then finding out that they have greatly over stated the potential of their methods. So if you have experience with the methods you are suggesting it will be even more awesome.
Here's a good quick introduction: intro to neural networks. Note that R has neural-network functionality, so no need to spend any time implementing NN yourself until you've given it a spin and decided it looks promising for your application.
Neural networks are not obsolete, but they have gone through a couple of hype cycles, and then after realizing they don't do everything as was claimed, their reputation goes into a trough for a while (we're currently in one of those). Neural networks are good at certain tasks, and generally are better for tasks in which a human can do a similar task, but cannot explain exactly how they do it.
Neural networks do not give you much insight into the system you're using them to analyze, even after they are trained and operating well. That is, they can predict what will happen (for some systems), but not tell you why. In some cases, that is fine. In others, that is not fine. Generally, if you want or especially if you already have an understanding of the rules of how something works, you can use other techniques.
But, for certain tasks, they work well.
For time-series in particular, see this question's discussion: Proper way of using recurrent neural network for time series analysis
While it is focussed on statistical pattern recognition, rather than time series forecasting, I would strongly recommend Chris Bishop's book Neural Networks for Pattern Recognition becuase it is the best introduction to neural networks in general, and I think it would be a good idea to get to grips with the potential pitfalls in the use of neural networks in a more simple context, where the problems are more easily visualised an understood. Then move on to the book on recurrent neural networks by Mandic and Chambers. The bishop book is a classic, nobody should use neural nets for anything until they feel confident that they understand the material contained in that book; ANN make it all too easy to shoot yourself in the foot!
I also disagree with mbq, nn are not obsolete, while many problems are better solved with linear models or more modern machine learning techniques (e.g. kernel methods), there are some problems where they work well and other methods don't. It is still a tool that should be in our toolboxes.