I am setting up an open source project on Github and want to make use of a GPL v3 project residing in Google Code repo.
How would I do that? Checkout the Google Code repo and make it (unmodified) available in my repo?
No. Don't do that. Simply document it as a dependency, optionally offer automatic downloading and building in your build system, and that's it. If you include third-party libraries in your distribution, most Linux distributions are going to hate you.