Easy way to re-organize images???

Over the years I have tried various different plugins to display images on my WP blog. That and changing from allowing WP to organize uploads and back again means that I actually have image files scattered all over my install. They all work, because the locations haven’t changed, but I plan to migrate to a new server soon so I am trying to get everything tidied up as much as possible.

Is there some clever and automated way to move the images and update the links to them at the same time?

Thanks

Jeremy