How to Save Incoming Backgrounds in Thunderbird

I used to share a convoluted way of doing this that involved plucking the jpg code out of the mail file, with the caveat "if someone knows an easier way, tell me about it!"

Well someone just did, by email. 

While viewing the message, go to View - Message Body As - Plain text. 

The HTML message will appear in plain-text, and all embedded images (including background) will become attachments.  You can right-click and save the attachment you're interested in! 

Then to restore the HTML go to View - Message Body As - Original HTML

Wow!  Thanks very much to Łukasz for the info!