<?php
$args=array(
'orderby' => 'name',
'order' => 'ASC',
)
$count=1; // A $count variable
?>
<?php foreach (get_categories( $args ) as $cat) :
if($count%3==0) // This condition will be true for 3,6,9,12..........
{
//Your Code ---
}
$count++ // Increment $count variable
?>
<h3><a href="https://wordpress.stackexchange.com/questions/246349/<?php echo get_category_link($cat->term_id); ?>"><?php echo $cat->cat_name; ?></a></h3>
<a href="https://wordpress.stackexchange.com/questions/246349/<?php echo get_category_link($cat->term_id); ?>"><img src="<?php echo z_taxonomy_image_url($cat->term_id); ?>" /></a>
<?php endforeach; ?>
I have added comments to understand.