The Irish song GIVE ME YOUR HAND / TABHAIR DOM DO LÁMH has lyrics.

How 'bout ONE MORNING IN MAY (The Nightingale Song)?

Jean Ritchie does a version of THE CUCKOO which could be metered down to a waltz.
There are actually two versions of this that could be waltzed to.

I don't know if there are lyrics to A GHAOTH ANDEAS / SOUTH WIND but it's a pretty tune.

I think "The Rose of Tralee" is 3/4.

Chauncy Olcott's MY WILD IRISH ROSE and he may have written the lyrics for WHEN IRISH EYES ARE SMILING.