get the child categories, using [get_categories()][1];
then loop through them with a foreach loop, using [WP_Query()][2] :


    <?php  $cats = get_categories('child_of='.get_query_var('cat')); 
      
    	foreach ($cats as $cat) :
    
    	$args = array(
    	'posts_per_page' => 3, // max number of post per category
    	'category__in' => array($cat->term_id)
    	);
    	$my_query = new WP_Query($args); 
    
    		if ($my_query->have_posts()) : 
    		echo '<h3>'.$cat->name.'</h3>';
    	 
    		while ($my_query->have_posts()) : $my_query->the_post(); ?>		
    		<?php /*general loop output; for instance: */ ?>
    		<a href="<?php the_permalink() ?>"><?php the_title(); ?></a>	<br />	
    	
    		<?php endwhile; ?>
    		
    		<?php else : 
    		echo 'No Posts for '.$cat->name;				
    		endif; 
    	 
    	wp_reset_query(); 
    	
    	endforeach; ?>

  [1]: http://codex.wordpress.org/Function_Reference/get_categories
  [2]: http://codex.wordpress.org/Function_Reference/WP_Query