Chad,
Depending on the Interact rules you are using (if using Interact) or the way the cases are created, you can make a simple Workflow rule that will send you an email when a Case is created and to avoid receiving an email for all Cases created, you can specify the Criteria to include logic that will be based on specific customers or any field value for that matter.
For further clarification, we suggest you register for one of our Advanced Q&A webinars (see the link-> http://www.clarizen.com/resources/online-demo.html#second ).These are great resources for free consulting with our Customer Success Managers and visual answers to your questions.
Sincerely,
Boris Krutiy - Clarizen Success