Extending WP_List_Table / handling checkbox options in plugin administration

I’m working on a WordPress plugin, and part of that plugin requires extending WP_List_Table and storing any of the items which are checked in that table to an option. I’ve managed to figure out how to properly setup and display the required table, but how do I handle storing the checked options?

Here’s what I’ve got so far…

class TDBar_List_Table extends WP_List_Table {

    // Reference parent constructor
    function __construct() {
        global $status, $page;

        // Set defaults
        parent::__construct( array(
            'singular' => 'theme',
            'plural' => 'themes',
            'ajax' => false
        ));
    }

    // Set table classes
    function get_table_classes() {
        return array('widefat', 'wp-list-table', 'themes');
    }

    // Setup default column
    function column_default($item, $column_name) {
        switch($column_name) {
            case 'Title':
            case 'URI':
            case'Description':
                return $item[$column_name];
            default:
                return print_r($item, true);
        }
    }

    // Displaying checkboxes!
    function column_cb($item) {
        return sprintf(
            '<input type="checkbox" name="%1$s" id="%2$s" value="checked" />',
            //$this->_args['singular'],
            $item['Stylesheet'] . '_status',
            $item['Stylesheet'] . '_status'
        );
    }

    // Display theme title
    function column_title($item) {
        return sprintf(
            '<strong>%1$s</strong>',
            $item['Title']
        );
    }

    // Display theme preview
    function column_preview($item) {
        if (file_exists(get_theme_root() . '/' . $item['Stylesheet'] . '/screenshot.png')) {
            $preview = get_theme_root_uri() . '/' . $item['Stylesheet'] . '/screenshot.png';
        } else {
            $preview = '';
        }
            return sprintf(
                '<a href="%1$s" class="thickbox" title="%2$s"><img src="%3$s" style="width: 150px;" /></a>',
                $preview,
                $item['Title'],
                $preview
            );
    }

    // Display theme description
    function column_description($item) {
        if (isset($item['Version'])) {
            $version = 'Version ' . $item['Version'];
            if (isset($item['Author']) || isset($item['URI']))
                $version .= '&nbsp;|&nbsp;';
        } else {
            $version = '';
        }
        if (isset($item['Author'])) {
            $author = 'By ' . $item['Author'];
            if (isset($item['URI']))
                $author .= '&nbsp;|&nbsp;';
        } else {
            $author = '';
        }
        if (isset($item['URI'])) {
            $uri = $item['URI'];
        } else {
            $uri = '';
        }

        return sprintf(
            '<div class="theme-description"><p>%1$s</p></div><div class="second theme-version-author-uri">%2$s%3$s%4$s',
            $item['Description'],
            $version,
            $author,
            $uri
        );
    }

    // Setup columns
    function get_columns() {
        $columns = array(
            'cb' => '<input type="checkbox" />',
            'title' => 'Theme',
            'preview' => 'Preview',
            'description' => 'Description'
        );
        return $columns;
    }

    // Make title column sortable
    function get_sortable_columns() {
        $sortable_columns = array(
            'title' => array('Title', true)
        );
        return $sortable_columns;
    }

    // Setup bulk actions
    function get_bulk_actions() {
        $actions = array(
            'update' => 'Update'
        );
        return $actions;
    }

    // Handle bulk actions
    function process_bulk_action() {
        // Define our data source
        if (defined('WP_ALLOW_MULTISITE') && WP_ALLOW_MULTISITE == true) {
            $themes = get_allowed_themes();
        } else {
            $themes = get_themes();
        }

        if ('update' === $this->current_action()) {
            foreach ($themes as $theme) {
                if ($theme['Stylesheet'] . '_status' == 'checked') {
                    // Do stuff - here's the problem
                }
            }
        }
    }

    // Handle data preparation
    function prepare_items() {

        // How many records per page?
        $per_page = 10;

        // Define column headers
        $columns = $this->get_columns();
        $hidden = array();
        $sortable = $this->get_sortable_columns();

        // Build the array
        $this->_column_headers = array($columns, $hidden, $sortable);

        // Pass off bulk action
        $this->process_bulk_action();

        // Define our data source
        if (defined('WP_ALLOW_MULTISITE') && WP_ALLOW_MULTISITE == true) {
            $themes = get_allowed_themes();
        } else {
            $themes = get_themes();
        }

        // Handle sorting
        function usort_reorder($a,$b) {
            $orderby = (!empty($_REQUEST['orderby'])) ? $_REQUEST['orderby'] : 'Title';
            $order = (!empty($_REQUEST['order'])) ? $_REQUEST['order'] : 'asc';
            $result = strcmp($a[$orderby], $b[$orderby]);
            return ($order === 'asc') ? $result : -$result;
        }
        usort($themes, 'usort_reorder');

        //MAIN STUFF HERE
        //for ($i = 0; i < count($themes); $i++) {

        //}

        // Figure out the current page and how many items there are
        $current_page = $this->get_pagenum();
        $total_items = count($themes);

        // Only show the current page
        $themes = array_slice($themes,(($current_page-1)*$per_page),$per_page);

        // Display sorted data
        $this->items = $themes;

        // Register pagination options
        $this->set_pagination_args( array(
            'total_items' => $total_items,
            'per_page' => $per_page,
            'total_pages' => ceil($total_items/$per_page)
        ));
    }
}

Problem is, I can’t get it to save properly. I select the rows I want, hit save and it just resets.