I found the easiest way was to do this:
<?php
$region = block_get_blocks_by_region('region_name_here');
print render($region);
?>
See the docs for more information: https://api.drupal.org/api/drupal/modules!block!block.module/function/block_get_blocks_by_region/7
In your module where you tried to print the sidebar region using the method used in a page template, use this snippet above instead. This method can also be used in html.tpl.php where the "usual" method cannot.