have a look at this Mudcat thread on medieval music - there are links that might provide some songs as well as music.
And just yesterday there was a programme on BBC R4extra that stated that Greensleaves was a tune known before Henry 8 and more than one lyric, even thought he may have written (or "borrowed" some of) the one we know best.
But be careful how you define medieval - see the Wiki link for some guidance.
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sj