To log in, you type in the e-mail. The server sends an email with a link (with secret code), which in turn sets a cookie when visited. The cookie expires in a month or so, and gets refreshed with every visit to the site. So the user only goes through the login process on a new computer.
One problem with this: it's too different from what's in place now. So users will be confused, at least initially. But this is actually an interesting idea.