The recitation about Rex, the piddlin' dog appears (along with a number of other nice recitations) on a recent CD by Irish flute player Kevin Henry. I think the CD is titled "One's Own Place".

As I recall (it's been a little while since the last time I listened to it), Mr. Henry's version has a different punch line, something like the town dogs wondering "what was the matter", but revealing that Rex "had a king-size bladder" or something very like that.