Shinedown/Tear Jerker

This post-grunge band has some incredibly moving songs.

"Some Day".
By far the most tearjerking song from them might be "What a Shame", particularly because it's about a man who had a drinking problem -- but nonetheless was a great person, a great husband, and a great father who would give you the shirt of his back and an all around nice guy. However, he was constantly judged by other people just because they couldn't see past his addiction.
"The Crow & the Butterfly" is a pretty tear-inducing song about a mother who lost her son and is trying to move on with her life, as difficult as it is. "I painted your room at midnight, so I'd know yesterday was over..." Although the song can have different representations, all of them deal with loss and the need of moving on, even if it's painful.
"Call Me" deserves a mention, as it has the potential to reduce you to tears on the first listen - if only because it pretty much come out of nowhere on their parent albums: after 9 (or 10 on Shinedown's album if you are American) songs of rock/metal riffage, the first thing you hear is a piano played with such emotion that you instantly feel your heart strings being tugged at and it gets more heart wrenching from there.
"Second Chance" can qualify, since it's a song that singer Brent Smith wrote about his relationship with his parents. In particular, his parents' difficulty in accepting that he is now grown up and independent.
- Not to mention the music video itself, which follows the story of a young woman who leaves her family to pursue her dreams. The last few shots are her surrounded by strangers on a bus as her family finds the note she left behind.
"Bully" can also be this -- especially for anyone who is being bullied, has been bullied, or knows someone who is/was bullied.
"45" is another one. It's a song about desperation and feeling weary with life.
"If You Only Knew" can be pretty sad on its own, but if you've just broken up with or lost someone you love...