How can I show recent posts from same taxonomy as the post currently being viewe

I’m wondering how I can show recent posts from the same taxonomy as the post that’s currently being viewed (working with custom post types and custom taxonomies).

If it was simply a category of a regular post, it would look like this:

<?php global $post;
$categories = get_the_category();
foreach ($categories as $category) :
?>
<h3>More News From This Category</h3>
<ul>
<?php
$posts = get_posts('numberposts=20&category='. $category->term_id);
foreach($posts as $post) :
?>
<li><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></li>
<?php endforeach; ?>
<li><strong><a href="<?php echo get_category_link($category->term_id);?>" title="View all posts filed under <?php echo $category->name; ?>">ARCHIVE FOR '<?php echo $category->name; ?>' CATEGORY &raquo;</a></strong></li>
<?php endforeach; ?>
</ul>

(found the above searching these forums)
—
How would I go about customizing this code for a situation that involves custom post types and custom taxonomies ?