Yet another reason to love Open Source! I adapted the Lifestream plugin by Chris J Davis and Drunken Monkey Labs, to create my links page. I give it the URL for the feed for sites I link to, and it retrieves the most recent post for each of them. It's, at best, a beta version, because of my "brute force coding" style. I really don't understand exactly how everything works. But it does work!
You can download it here. A quick note, the theme linkstream.php file will probably need some adjustments to look decent in another theme.