Arduinme:
I'm having some trouble using it to send SMTP mail even though similar code works fine on my Ethernet shield.
If it's similar code, that might be the problem. The Yun (and the clone shields) are completely different from an Ethernet Shield. Any code written for the Ethernet Shield and it's libraries will not run on a Yun. It's an almost total rewrite.
It's easiest to do all if the communications through the Linux side, especially sending mail. But if you want to do it from the sketch, you will have no luck if the sketch includes Ethernet.h - you need to go through the Bridge library.