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

Have you tried using get_the_terms()?

Quick-and-dirty, from your code example:

<?php 
global $post;
$terms = get_the_terms( $post->ID, 'some-term' );
foreach ($terms as $category) :
?>
<h3>More News From This Category</h3>
<ul>
<?php
$posts = get_posts('numberposts=20&category='. $category->term_id);
foreach($posts as $post) :
?>
<li><a href="https://wordpress.stackexchange.com/questions/39455/<?php the_permalink(); ?>"><?php the_title(); ?></a></li>
<?php endforeach; ?>
<li><strong><a href="<?php echo get_category_link($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>

See also: the_terms() and get_the_term_list()