The voice of Grinder the bulldog is played by Greg Morton, who has also been a voice actor on the Super Mario Bros. Super Show and "Captain N/Super Mario Bros. 3", in addition to playing Jamie King on "Scarecrow and Mrs. King".
Another possibility is Will Ryan who is the voice of Willie the Giant (a.k.a. the Ghost of Xmas Past in "Mickey's Christmas Carol"), but he has been more well-known for roles in Disney features.
However, Will Ryan's name is not listed on the back of "Hello Kitty's Furry Tale Theatre", so Greg Morton seems more logical, even though Will Ryan's portrayal of Willie the Giant sounds similar to Grinder's character voice.