Showing post thumbnail (attachment) on the archive.php category listing

You could use the WordPress function get_children. Although I don’t think it makes a difference, performance-wise.

<?php while (have_posts()) : the_post(); ?>
    <?php $attachment = array_values( get_children( array( 'post_parent' => $post->ID, 'post_type' => 'attachment', 'post_mime_type' => 'image', 'order' => 'ASC', 'numberposts'  => 1 ) ) ); ?>
    <div class="searchItem" style="clear:both;">
        <h3 id="post-<?php the_ID(); ?>">
        <?php if( $attachment ) echo '<img src="' . wp_get_attachment_url($attachment[0]->ID) . '" class="post-attachment" />'; ?>
        <a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h3>
        <small><?php the_time('l, F jS, Y') ?></small>
        <div class="excerpt"><?php echo $post->post_excerpt; ?></div>
        <div class="postmetadata">Posted in <?php the_category(', ') ?> | <?php comments_popup_link('No Comments &#187;', '1 Comment &#187;', '% Comments &#187;'); ?></div>
    </div>
<?php endwhile; ?>