The Walker_Category class is what ultimately worked for me. It had all the of the leverage I needed to make a custom list.
However, I just wanted to post my first attempt, that worked pretty well for a simple list. It uses nested loops.
<div class="tab-row">
<?php $args = array(
'taxonomy' => 'taxonomyName',
'parent' => 0
);
$categories = get_categories($args);
$catid = array();
foreach($categories as $category) {
echo '<ul class="parent-tab"><div class="parent-item">' . $category->name . '</div>';
array_push($catid, $category->term_id);
echo '</ul>';
} ?>
</div>
<div class="child-row">
<?php
$countStop = count($categories);
$i = 0;
while ($i < $countStop) {
echo "<ul class='child-list'>";
$args = array(
'taxonomy' => 'taxonomyName',
'parent' => $catid[$i]
);
$categories = get_categories($args);
foreach($categories as $category) {
echo '<li class="child-item">' . $category->name .'</li>';
}
echo "</ul>";
$i++;
}
?>
</div>
I used get_categories() which eliminated the links, and created my taxonomy hierarchy by only displaying the parents in the first loop, and in the nested loop, pulling out the children of that parent by retrieving the parent ID.