Flogging Molly/Tear Jerker

This Celtic punk band can trigger the tears.

"Don't Let Me Die (Still Wondering)". This song was was allegedly written as a tribute to Johnny Cash -- when the band's lead singer, Dave King, heard of Cash's death/
"The Spoken Wheel" is a song written for King's late father. It also becomes a Crowning Moment of Awesome -- when it leads into the following track: "With a Wonder and a Wild Desire", which also has a certain furious defiance that can inspire a few Manly Tears.
Some of their more punk-orientated songs, such as the tragic "Laura" and the bittersweet "Within a Mile of Home", are also well worth a mention.
"Float". Especially with the gorgeous but heartachingly sad video.
"Tobacco Island" is also particularly sad, which is about the Irish being thrown out by Oliver Cromwell, and sent to the Carribean. The song is about the singer remembering and mourning the loss of his homeland.
- In particular, the bridge of the song; Dave King sings as the narrator with such a sense of heartbreak in his voice, you can't help but cry. The lyrics:

Agony, will you cleanse this misery?
For it's never again i'll breathe
The air of home
From this sandy edge
The rolling sea breaks my revenge
With each whisper a thousand waves
I hear roar
I'm coming home

"A Prayer for me in Silence".
"What's Left of the Flag".