Conflict Minerals in Electronics: What to look out for as a designer?
This not legal advice, disclaimer, don't listen to anything I say, and definitely don't sue me, you should seek professional legal advice on the matter to make sure your liability is covered etc, etc :)  I'm far from an expert but here is some information that might help you on your way.

In the US at least your requirements and responsibilities as a public company are pretty well laid out in section 1502 of the Dodd-Frank bill.  Wiki has an overview and these guys also had a nice write up about it

In this pdf from the SEC you can see the materials mentioned like this:  

The term “conflict mineral” in the final rule is defined to include cassiterite, columbite-tantalite, gold, wolframite, and their derivatives, which are limited to the 3Ts, unless the Secretary of State determines that additional derivatives are financing conflict in the Covered Countries, in which case they are also considered “conflict minerals;” or any other minerals or their derivatives determined by the Secretary of State to be financing conflict in the Covered Countries. 

So your actual list is probably subject to change if all of a sudden mineral-X is being used to finance wars.

Now if you're not a public company, but you might supply one, you can bet eventually they will come to you and ask for assurances that your product is in compliance with the above regulation.   Probably right before they place that large order your counting on.  The standard mode of operation appears to be put downward pressure on your suppliers, asking for letters or certificates verifying that they comply.  Basically trying to cover your butt by putting the liability on suppliers to comply.  That doesn't mean that will save you if someone finds your supplier wasn't telling the truth...

What was interesting to me in our research was some suppliers had prepared letters, documentation etc, yet some very large capacitor suppliers responded that they could not verify that their material was not sourced from a conflict region.

The act itself requires companies to check country of origin in good faith, and also to disclose how they did it as well as being independently audited.  So any and all of that might be required of you by your customer.  It really expects you the company to take an active role in making sure you don't use this material, not just a hey I didn't "knowingly" do this wink, wink, nudge, nudge.

Hope that helps a little, again I'm far from an expert on the topic.