This was one of the fastest ones I wrote.  The idea for the "greyhound" rhymes popped into my head fairly quickly, aided by the lucky fact that the rhymes fit so well with the subject matter.  It only took a few more minutes to work out what the two shorter lines should be, and the whole thing fell into place.  I think the part that took the longest was coming up with the word "beguile" when I was looking for a two-syllable synonym for "catch."

An admirable dog is the greyhound.
It's an excellent sit-, heel- and stay-hound.
But should something dash by
And beguile its eye,
It's a blurred streak of running-away-hound.

Based on the suggestion: “greyhound”