I scoured CDDB. This is what I found. ANy comments on any of them? I haven't heard most of them, but I trust the bands.
"January Friend," The Goo Goo Dolls
"January" Goldfinger
"June in January" Mary Ford and Les Paul
"New Year's Day" U2 (yeah, I should have thought of this, but we're getting U2 heavy on my suggestion list)
"This New Year" Death Cab for Cutie
"New Year" The Breeders
My brother-in-law recorded a song about building a snowfort in January, maybe I'll try to get his permission.
"Xmas in February" Lou Reed
"Two Days in February" The Goo Goo Dolls
"February Stars" The Foo Fighters
"One February Friday" Frankie Goes to Hollywood
"February Moment" Herbie Hancock and Chick Corea
"February" Jesus Jones
"Ground Hog's Day" Primus
"The Ides of March" Iron Maiden
"Waters Of March" David Byrne and Marisa Monte (or Cassandra WIlson)
"St. Patrick's Day" John Mayer
There's a lot of stuff about Marching, but not much about the month of March
"April Fool" Soul Asylum (Ok, should have remembered this one)
"April Skies" The Jesus and Mary Chain
"April Fool" Pete Townshend and Ronnie Lane
Lot of songs for "may" as in "may I" but again, few months.
"Cinco de Mayo" Liz Phair
"May Queen" Liz Phair (she's a little weird)
"First of May" James Taylor
"Evening in June" Van Morrison
"Rainy Day in June" The Kinks (another one I should never have forgotten)
"53 Days In June" Phil Keaggy, Wes King, Scott Dente
"1000 Julys" Third Eye Blind
"July, July!" The Decemberists
"August 8th" NOFX
"Stuck in the August Rain" Jethro Tull
"Cold Wind in August" Van Morrison
"August and September" The The
"Wake Me When September Ends" Green Day
"September Grass" James Taylor
"September Skies" The Brian Setzer Orchestra
"September Night" Van MOrrison
"September Song" Ella Fitzgerald
"October Road" James Taylor
"October Winds" Béla Fleck
"Halloween" The Dead Kennedys
"Halloween" Sonic Youth
"Halloween" Siouxsie and the Banshees
"November Spawned a Monster" Morrissey
"November 5" Love Spit Love
"Thanksgiving in Reno" Too Much Joy
"Thanksgiving (No Way Out)" Violent Femmes
"A Long December" Counting Crows
"December" Weezer
"Forget December" Something Corporate
"December is for Cynics" The Matches
And of course a billion Christmas songs.
"What are You Doing New Year's Eve" Harry Connick Jr.