There are still talking animals in Dragon Ball Z, in the episode Plight of the children, the police officer is an animal, and some of the children are too.
Also, near the end of the Cell Saga, you had a cut to the King of the Earth (a dog), commenting on how much adult Goku, looked like the child Goku who killed King Piccolo.
But the number of talking animals did indeed get reduced.
It kinda goes with the direction Dragon Ball was taking. It was moving away from being a fanciful, light-hearted comedy to a more serious fighting shonen, and the talking animals fit less and less to the story.