"January Hymn" by The Decemberists
"February Stars" by Foo Fighters
"Winds of March" by Journey
"April Fool" by Manchester Orchestra
"Month of May" by Arcade Fire
"June" by Pete Yorn
"4th of July" by Soundgarden
"August" by Rilo Kiley
"Pale September" by Fiona Apple
"October" by Broken Bells
"Gone 'til November" by Wyclef Jean
"My December" by Linkin Park

Without looking any up, here's what I came up with:

January - ???
February - "February Air" - Lights
March - "Into the Mouth of Hell We March" - Trivium (Wrong context, same spelling )
April - ???
May - "Wherever I May Roam" - Metallica (Wrong context, same spelling )
June - ???
July - ???
August - "At the End of August" - 36 Crazyfists
September - "Wake Me Up When September Ends" - Green Day
October - ???
November - "November Rain" - Guns N' Roses
December - "My December" - Linkin Park (yeah, I know you already covered that one )

Here's my take on this one... I could have used Barbara Dickson for either January or February, but she was the only February in my database whereas I had others for January!

The Month of January - June Tabor
January February - Barbara Dickson
Waters of March - Jane Monheit
April Come She Will - Simon & Garfunkel
First of May - The Bee Gees
June Afternoon - Roxette
Cold Day in July - Suzy Bogguss
It Was a Beautiful Day in August - Diana Krall
It Might as Well Rain Until September - Carole King
October - U2
November - Tom Waits
December Will Be Magic Again - Kate Bush