How to show previous and next post excerpts?

I’m looking for a way to show excerpts for the previous and next post. There doesn’t seem to be a way to do this with a simple tag, like previous_post_link.

I want to be able to have complete freedom to style the excerpt or to use other code in the previous and next post content, so I’d prefer to stay away from a plugin. It seems like it should be fairly easy using multiple loops, but I can’t figure it out. What code do I need to pull the excerpt?