In the Yeats poem, "The Ballad of Father Gilligan," the fact that half of his flock, half of his parishioners, are lying under green sod is responsible for the priest's exhausted state.  The green sod refers to their graves. 

Half of his flock is dead, and the father can't keep up.  His job is to comfort the dying and administer the last rites.  His people are dying so fast that he is exhausted.  Even as he dozes off in a chair, he is summoned by another man dying.   

The image of the green sod and the graves of his parishioners contributes to the plot of the narrative, explaining and echoing the death of the man who sends to him, as well as explaining why the priest falls asleep and fails to respond to the man's call.  This sets up the intervention of God, who sends an angel to take the poor priest's place at the dying man's side. 

Although in British English the word "sods" can be an obscenity, it also has a "decent" meaning.  Sod in that context is an area of grass that also includes the dirt beneath it that is held together by the roots of the grass.

So, if part of his flock is under areas of grass like this, where would they be?  People who are under the grass are dead people who have been buried.

So half of his flock lies under the green sods because they are dead.  Gilligan is old and many of his people have died.