I have, in my naivety, set up a git submodule and treated it like a Subversion external - i.e. it's now full of changes that I've just realized haven't been committed or pushed anywhere.

Is there some easy way to commit/push the submodule changes back to the upstream repo? And what's the recommended technique in Git for doing simultaneous development on separate (but linked) repositories in this way?

A submodule is its own repo/work-area, with its own .git directory.

So, first commit/push your submodule's changes:

$ cd path/to/submodule
$ git add <stuff>
$ git commit -m "comment"
$ git push

Then, update your main project to track the updated version of the submodule:

$ cd /main/project
$ git add path/to/submodule
$ git commit -m "updated my submodule"
$ git push