Thanks (and that was very fast typing! ;)
1) Create a new Moodle site:
If "but it may carry with the legacy files" materializes, my the whole exercise would be useless.
2) Manually delete all of the legacy files:
That is what I thought. Simply make the moodledata/courseno/ directories unreadable to the web-server (for later removal). Wanted opinions before I send the teachers looking for missing files!
3) Rebuild your courses manually in the 3.1 installation:
Out of the question. (It is not _me_, you know. ;