"Down by the Salley Gardens" sounds like a good Irish folk song because, well, it was inspired by one. So Yeats wasn't trying to make his rendition sound too different in terms of its sonic qualities, nor was he trying to throw a bunch of fancy technical curves at the reader.
We talk about the poem's rhyme scheme and meter in "Form and Meter," so check out that section for the goods on how the poem is put together. As far as the sounds of the poem itself, the word choices here don't add up to much in the way of alliteration, assonance, or consonance. Instead, it all sounds rather simple and straightforward as the speaker tells his familiar story about young love, heartbreak, and the advice he should've taken. No special sound tricks here—instead, we get just plain, old-fashioned romantic heartache and mega-frownie faces.