Try out the "Capella de Ministrers", a chamber of Medieval music re-enactors, or the Cantigas de Santa Maria, musicated by kinf Alfonso the Wise, of Castille. Songs in old Castillian and Occitan. You can't have a true Medieval mod without songs in Provençal, the language of troubadours and minstrils.
From the Cantigas:
http://www.youtube.com/watch?v=Z1klIcr0d5E
Try out several troubadours, like:
Raimbaut de Vaqueiras
http://www.youtube.com/watch?v=KMiNOMEvsAs
http://www.youtube.com/watch?v=MZpuzb6cbJU
http://www.youtube.com/watch?v=5CHFLYhmqxA
Beatriu de Dia (a trobairitz, a woman troubadour):
http://www.youtube.com/watch?v=Dt8dWGH7nf0&feature=related
Berenguer de Palou:
http://www.youtube.com/watch?v=xkh6LhWj5IA&feature=related
http://www.youtube.com/watch?v=pxmpfvvGHMA&feature=related
And many others. Peire Vidal, Jofre de Foixà, Guillem de Berguedà, Guillem de Cabestany... Check out Wikipedia's list for more troubadours, many of them have some songs which survived till today and mostly they wrote them between 1150 and 1300:
http://en.wikipedia.org/wiki/Category:Troubadours
The worst name for a Medieval music re-enactor group: Mediaeval Baebes. But they have some interesting songs
http://www.youtube.com/watch?v=dVKLo9YepUg
What about Scarborough Fair? Isn't it from the XIVth Century? (yes, the version could be better, but the only one besides this I found was Simon & Garfunkel's...)
http://www.youtube.com/watch?v=2lTDDf9NoLg
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Funny thing my language is so close to old Occitan, I can understand most of those songs.
By the way, Thibaut's song sounds a lot like Occitan. Curious how pronunciation changed through the ages, although English is the master of pronunciation mutations with its Great Vowel Shift.