I came up with a solution for this:
$fields = get_fields();
if( $fields )
{
// Reorder based on menu order, rather than id
foreach( $fields as $field_name => $value )
{
$field = get_field_object($field_name, false, array('load_value' => false));
$menu_order[$field_name] = $field["menu_order"];
}
array_multisort($menu_order, SORT_ASC, $fields);
}