On the SmashinGeeks.com site there's a recent post showing you how to make a simple request and get the number of retweets for a post using PHP and Tweetmeme/Backtype.

Tweeting is what we do after publishing posts. To place some minimal text that jumps upon the Tweet on Twitter will be more great instead of using a tweet button. Its made possible to count the number of tweets using PHP only, however if you want to sizzle it up, then its your way.

They include code examples of how to connect to each of the services and make a request for a query string to their API using libraries provided by the service.