Problems in setting up a Custom Taxonomy Term template

I have a Custom Post Type and 2 taxonomies attached to it. Each taxonomy has about 20 terms containing several posts. While I have successfully setup a custom archive template <archive-mycpt.php> for my custom-post-type. My Taxonomy terms are creating problem. When I view any of the terms page (by clicking them from CPT menu) it shows me the titles of the posts under it.

I tried setting up taxonomy-my-taxonomy-my-term.php but the page does not pick it up. Instead it is using archive template. Here is the body class that is returned when i view the term page where <a-path> is the term:

archive tax-path term-a-path term-19 logged-in

Which template do I need to make in order to list all posts under a specific term, and can I have a generic template for all terms, since all of them have posts with similar data fields.