Using Github's Issue Tracker, how can one create an issue tied to an organization (and not just a specific repository)?
For example, the issue might affect several repositories or the company as a whole, or the person creating the issue might not know which repository it affects.
One hackish way is to have a catch-all/generic repository where people put the general/company-wide issues, but is that the only way? (ideally this could be done in github and not in another site/service since the overall needs are not much).
Yup, creating a generic repo and storing the issues there is the only way. There's no way to create an issue for an entire organization.