Can I make unlocking the screensaver also unlock the login keychain?
This usually happens when you change your login password without changing your keychain password.
- Open Keychain Access
- Click Edit -> Change Password for Keychain 'Login'
- Match the password to the one you log into the computer with.
If you firmly believe "this is not the problem", please try to delete your keychain and recreate it in order to get the link between the login and keychain to properly function. (thanks @dragon788)