Although move notifications could be easily disabled, some of the other email notifications could not. Although greatly reduced anyway by the site notification system that replaced many emails, the remaining few can be disabled altogether in "Site settings >> Notification Settings".
The process for turning off email notifications has been overhauled.
Every "notification" email now has an unsubscribe link in the footer.
These are "Move Notifications", "Opponent Initiated Move reminders" (rarely used these days), and "Everything else" (open invites accepted, game deleted, etc)"
Clicking the the "unsubscribe from email" link will stop similar emails (as defined above) from being sent. You no longer need to be logged for the link to work.
Email notifications can be turned back on in Accounts Settings >> Notifications.