Search cases - find case by case role
Try the below steps-

1) Navigate to Search -> Advanced Search.

2) In Basic Criteria Tab, you can find a dropdown with the label Display Results as -> Select Cases in the dropdown input.

3) In Relationship Tab select the relationship type as the role. For eg.- Case coordinator is.

4) Optionally type the name in the Target contact textbox. 

Hit Search -> Result shows the case list supervised by the contact.

Change the Display Results as to Contacts etc as per your requirement for the returned result.