Using print render($content['taxonomy']); in node.tpl.php
I think the proper way to do this would be to implement THEME_preprocess_HOOK() in your theme's template.php.  

In that function, just add each taxonomy term name into the classes array.  You'll probably also want to cleanse the term names as you put them into the classes array, dig around Pathauto or Transliteration modules to find a function to do the dirty work.

function mytheme_preprocess_node(&amp;$vars) {
    foreach ($vars['node']-&gt;field_my_taxonomy_field[LANGUAGE_NONE] as $term) {
        $vars['classes_array'][] = pathauto_cleanstring($term-&gt;name);
    }
}