Certainly signing your requests as actually having come from your application is a legitimate means of security. But there are much lower hanging fruit (in terms of available security measures). Specifically, using SSL to handle API traffic. That should absolutely always be step 0. In instance of this specific post, had the API provider enforced using SSL as their transport protocol we probably wouldn't even be having this discussion.
There's a ton of other security measures/protocols that should be taken and followed, most of which are already talked about in other parts of these comment threads. I just wanted to really point out that what you're describing above sounds an awful lot like an HMAC code.
[1] http://en.wikipedia.org/wiki/Hash-based_message_authenticati...