Moving thumbnails to subdirectory in uploads folder

Is it possible to move all generated thumbnails to a subfolder within wp-content/uploads, for example leave original images in wp-content/uploads but have thumbnails in wp-content/uploads/thumbs.

If you’re asking yourself why, I’m having a problem with Google indexing and showing image thumbnails in results rather than full-sized images, so I’d like to noindex the /thumbs folder.