Who wrote (or said): "Grieve not...nor speak of me with tears...but
laugh and talk of me as though I were beside you. I loved you
so...'twas heaven here with you."

Hi! Thanks for the question.

The writer who wrote such words was Isla Pachal Richardson. The eulogy
seems to be popular in dog websites.

The following links give citations to such poem.

"Rainbow"
http://www.suzyqranch.com/rainbow.html

"Sheltie Angels Among Us - Loving Words Page Two"
http://www.aileashelties.com/words-page-02.htm

"The Staffordshire Bull Terrier Rainbow Bridge - They Still Live"
http://www.internationalstafford.com/mag2000/Articles/rainbowbridge/stilllive.html

Search terms used:       
"Grieve not" "nor speak of me with tears"
  
I hope these links would help you in your research. Before rating this
answer, please ask for a clarification if you have a question or if
you would need further information.
      
Thanks for visiting us.      
      
Regards,      
Easterangel-ga      
Google Answers Researcher

Thanks for the prompt answer. By using the entire phrase as a search
term on Google, I discovered the text of the full poem ("To Those I
Love") by Isla Paschal Richardson.