Rom this is a very famous and beautiful poem by Mary Elizabeth Frye.
It may not be about a dog but I think it fits
I Am Not There
Do not stand at my grave and weep; I am not there. I do not sleep.
I am a thousand winds that blow, I am the diamond glints on snow.
I am the sunlight on ripened grain, I am the gentle autumn's rain.
When you awaken in the morning's hush,
I am the swift uplifting rush of quiet birds in circled flight.
I am the soft stars that shine at night.
Do not stand at my grave and cry;
I am not there:
I did not die.
It really means that your memories of him will live on and always be with you and you will see him in the common things of life.