Hi Martin,
How are you setting up this customer? Do you have them created in a Customer "group" or do you have the users in a User Group? What you could do is to turn off each of the user's settings the user's "My Details > Project Notifications" and "Notify me" to "None". If you have a lot of users, you can streamline this by creating a workflow rule if you have a User Group with the customer's users and basically set these settings automatically once they join the group.