använd LEFT JOIN
SELECT a.name
FROM staff a
INNER JOIN staff_prospects b
ON b.staff_id = a.id
LEFT JOIN staff_prospect_activity c
ON c.prospect_id = b.id AND
c.date_of_activity >= '2013-05-31' AND
c.date_of_activity < '2013-05-31' + INTERVAL 1 DAY
WHERE c.prospect_id IS NULL
GROUP BY a.name