In the end I used sSMTP. It's far simpler than either Postfix or sendmail and does the job beautifully.

For future reference, here's how to use sSMTP with Yahoo Mail (don't worry, it's a lot less complex than it looks):

• Use your package manager to download ssmtp. I used Synaptic, but sudo apt-get install ssmtp should work as well.

  
  



• Open the config file at /etc/ssmtp/ssmtp.conf.

  
  



• Make the config look like this:

  
  

  [email protected]       <--- Use your real email here
  
  mailhub=smtp.mail.yahoo.com:587       <--- This will depend on your email provider
  
  FromLineOverride=YES
  
  UseSTARTTLS=YES
  
  [email protected]   <--- Use your real email here
  
  AuthPass=yourRealYahooPassword        <--- Use your real email password here
  
  TLS_CA_File=~/cert.pem
  
  

  
  



• Create the cert.pem file with OpenSSL. I used openssl req -x509 -newkey rsa:2048 -keyout key.pem -out cert.pem -days 9999 -nodes (more info at How to create a self-signed certificate with OpenSSL). You can stick the file anywhere, but I just chucked it in ~/. Wherever you put it, make sure you point the TLS_CA_File= line in ssmtp.conf to the correct location.

  
  



• Open the file /etc/ssmtp/revaliases and add the line (replacing the appropriate sections with your details):

  
  

  yourPCUsername:[email protected]:smtp.mail.yahoo.com:587
  
  

  
  

  If you're running as root, I assume you would need to add another line replacing your name with 'root'.

  
  



• That's it, you're good to go! To test, the easiest way (IMO) is to create a file with the following in it:

  
  

  To: [email protected]                       <--- Use your real email here...
  
  From: "whateverYaWant" <[email protected]>  <--- ...and here
  
  Subject: Some Notifying Email
  
  MIME-Version: 1.0
  
  Content-Type: text/plain
  
  
  
  Body of your email goes here. Hello world!
  
  

  
  



• Check that you don't have the real sendmail installed by running sendmail -V. It should say 'sSMTP'.

  
  



• Finally, run cat fileWithEmailInIt.txt | sendmail -i -t, then wait a few seconds (10-30) and check your email!

  
  

Additional note 1: If you have trouble with Gmail, try option 1 of this answer.

(Thanks to Ben Creasy!)

Additional note 2: If mail is sending from the command line but not through crontab, try changing FromLineOverride to NO in /etc/ssmtp/ssmtp.conf. You can also get more detailed logging by adding Debug=YES to ssmtp.conf - the extra logging goes to /var/log/mail.log.

(Thanks Jakub Kukul!)