The function you mention is going to return a list of links -- aren't you just trying to get the term slug as a class? Why not do....


	$terms = get_the_terms($post->ID, 'my-category');
	$slug_out = array();
	foreach($terms as $term){
		$slug_out[] = $term->slug;
	}
	
	echo '<div class="'.implode(', ', $slug_out).'">';