Suites (and the tests within them) may be transferred between organizations that you belong to, including to and from your "Personal" organization. This can be done from the suite screen by clicking the "Actions" menu and choosing the "Move to Organization..." option.
After selecting this option you will be presented with a modal dialog box with all the organizations that you belong to. Simply select the organization where you would like to move the suite and submit. The suite, along with all its tests, will be moved to that organization.