I'm surprised that in 9 pages of discussion these weren't mentioned once....
The Priest and The Matador - Senses Fail
Bittersweet - Apocalyptica
Bittersweet Symphony - The Verve
Does He Love You? - Rilo Kiley (surprisingly upbeat, very sad)
Is It Real? - The Seatbelts (From Cowboy Bebop)
Black Dahlia - Hollywood Undead (Seems a bit angry as well)
House Carpenter - Hurt
Losing - Hurt
Rapture - Hurt
Slow Dancing In A Burning Room - John Mayer
The Nurse Who Loved Me - A Perfect Circle
Uninvited - Alanis Morissette
I Don't Care - Apocalyptica
Desperately Wanting - Better Than Ezra
Change - Blind Melon
Mouth Full Of Cavities - Blind Melon
The Dress - Blonde Redhead
Cure My Tragedy (A Letter To God) - Cold
Insane - Cold
Rain Song - Cold
Into The Labyrinth - Dead Can Dance
(I have plenty more, and may add, depending on responses to the thread.)
I also agree with ones that were posted before:
Hurt - NIN
Almost Lover - A Fine Frenzy
Your Ex-Love Is Dead - Stars
Cats In The Cradle - Henry Chapin
But yes. I apologize for my ridiculously long list.