How do I rewrite this loop as a new WP_Query style-loop?

You’re 95% of the way there already. Just add if ( $my_query->have_posts() ) and incorporate your $odd variable:

<?php 
$my_query = new WP_Query('offset=5&showposts=10');

if ( $my_query->have_posts() ) :

    $odd = false;

    while ($my_query->have_posts()) : $my_query->the_post();

        $odd = !$odd;

        ?>

<div class="<?php if ($odd) echo 'uneven '; ?>post">

<h3><a href="https://wordpress.stackexchange.com/questions/70713/<?php the_permalink(); ?>"><?php the_title(); ?></a></h3>

<?php the_excerpt(); ?> 

</div>

    <?php

    endwhile;

endif;