Count all comments of a custom post type

How about direct sql with a simple subquery to first get all the post ids of your custom post type ex:

function get_all_comments_of_post_type($post_type){
  global $wpdb;
  $cc = $wpdb->get_var("SELECT COUNT(comment_ID)
    FROM $wpdb->comments
    WHERE comment_post_ID in (
      SELECT ID 
      FROM $wpdb->posts 
      WHERE post_type="$post_type" 
      AND post_status="publish")
    AND comment_approved = '1'
  ");
  return $cc;
}

Usage:

$total_comments =  get_all_comments_of_post_type('some_post_type');