Using a custom template in a theme I have created the custom template page-org.tpl.page in the Zen theme, and I want to include this custom template file in the menu calling; when the user clicks on the Org link it should use the template file page-org.tpl.page.
How can I do that?
Output your $vars in the theme so you can see your candidate templates.

EDIT: I thought you were looking for a content type solution when I first read this :)

Here's a gotcha: 
you have to have master templates before you can have ones within it. For example content-field-field_my_content.tpl.php will not work until content-field.tpl.php is in place in the theme.

You may need to place some PHP in your template.php within preprocess_page()

if(!arg(2)) $variables['template_files'][] = 'page-node_'.$variables['node']-&gt;type;

(The arg2 check helps make sure this doesn't happen on edit pages.)

Lastly, you can also accomplish this with Context Layout.