... And a checklist of songs from the 13th-15th centuries that will at least give you titles to search:

http://www.wsu.edu/~delahoyd/medieval/m.e.lyrics.html

Also, I don't know the date of origin of the following, but here are some medieval and renaissance song titles from a page that includes texts to all of them, plus music where known. Source:

http://www.luminarium.org/medlit/mltexts.htm

Fowles in the Frith
Alisoun
My Lief Is Faren in Londe
Western Wind
I Have a Yong Suster
The Cuckoo Song
Adam Lay Ybounden
I Sing of a Maiden
The Corpus Christi Carol
The Agincourt Carol
Bryd one Brere
Man Mai Longe Liues Wene
Lenten ys Come
When the Nyhtegale Singes
Blow, Northern Wind
Mery It Is
Man in the Moon
Nowell, Nowell, Nowell
Nowell Sing We
Nowell: Out of Your Sleep
Alleluia. Now well may we mirthës make
Good Day, My Lord Sire Christmas
Of A Rose Sing We
Ave Domina
Alma Redemptoris Mater
As I Lay Upon a Night
What Tidings Bringest Thou?
Sing We To This Merry Company
Sing we Yule
Man, Assay
I Pray You All
God Speed the Plow!
Women [warning: bawdy]
More Selected Medieval Lyrics