Gustav pointed me to SSHKeychain. I haven't tried it yet, but if it works then it's brilliant.
When I used Linux on my desktop I'd use ssh-agent and have it span all my windows to be able to login to various boxes without typing my password a billion times a day.
SSHKeychain is supposed to do the same thing across sessions on OS X by using the OS X Keychain system. Clever. It knows to lock the keychain and ask for the password again after the computer has been asleep or had the screensaver enabled. It requires me to logout and login again to work, so I haven't tried it yet, but it looks just like what I've been missing all along.