I was able to get it to work, using a Gmail account. Here are the relevant settings:
Mailer: Send all WordPress emails via SMTP.
SMTP Host: smtp.gmail.com
SMTP Port: 587
Encryption: Use TLS encryption.
Authentication: Yes: Use SMTP authentication.
Username: [your - entire - Gmail or Google Apps email address]
Password: [your password for your Gmail or Google Apps email account]