This ended up being a couple of extra steps beyond just Googling, so I wanted to put it out there for the next person.
I found these instructions on using the Gimp plugin Synthesizer to remove objects to be pretty useful, though it looks like some of the menu options have moved around in Gimp 2.8.
Getting the Synthesizer plugin installed correctly on Ubuntu took a couple of tries. Here are great instructions for adding a PPA to your config so you can just use a couple of apt-get commands to install Gimp and many useful plugins, including Synthesizer. It came down to this:
sudo add-apt-repository ppa:otto-
kesselgulasch/ gimp sudo apt-get update sudo apt-get install gimp sudo apt-get install gimp-plugin-registry