I have argued before that I really don't see why pet owners should not have the right to try to claim personal/emotional injuries due to the deaths or injuries to their pets. See here and here and there are a few cases out there that have recognized these types of claims, (See here for example.)
However courts have traditionally held that animals are "chattels" and that there is no cause of action for emotional distress, loss of companionship and the like when they are injured by others.
The most recent case on the subject comes from the New Jersey Supreme Court which has held that a plaintiff may not recover damages for the emotional injury caused by witnessing the death of her pet dog. Following the traditional analysis, the court held that a pet is merely "property."
The case is called McDougall v. Lamm. For more on the story go here.
UPDATE: Torts Today has more on the case here.