functions.php

<?php
/*
Description: Adds a taxonomy filter in the admin list page for a custom post type.
Written for: http://wordpress.stackexchange.com/posts/582/
By: Mike Schinkel - http://mikeschinkel.com/custom-workpress-plugins
Instructions: Put this code in your theme's functions.php file or inside your own plugin. Edit to suite your post types and taxonomies. Hope this helps...
*/
add_filter('manage_listing_posts_columns', 'add_businesses_column_to_listing_list');
function add_businesses_column_to_listing_list( $posts_columns ) {
	if (!isset($posts_columns['author'])) {
		$new_posts_columns = $posts_columns;
	} else {
		$new_posts_columns = array();
		$index = 0;
		foreach($posts_columns as $key => $posts_column) {
			if ($key=='author') {
			$new_posts_columns['businesses'] = null;
			}
			$new_posts_columns[$key] = $posts_column;
		}
	}
	$new_posts_columns['businesses'] = 'Businesses';
	return $new_posts_columns;
}

add_action('manage_posts_custom_column', 'show_businesses_column_for_listing_list',10,2);
function show_businesses_column_for_listing_list( $column_id,$post_id ) {
	global $typenow;
	if ($typenow=='listing') {
		$taxonomy = 'business';
		switch ($column_id) {
		case 'businesses':
			$businesses = get_the_terms($post_id,$taxonomy);
			if (is_array($businesses)) {
				foreach($businesses as $key => $business) {
					$edit_link = get_term_link($business,$taxonomy);
					$businesses[$key] = '<a href="'.$edit_link.'">' . $business->name . '</a>';
				}
				echo implode(' | ',$businesses);
			}
			break;
		}
	}
}


/* JUST AN HYPOTHETICAL EXAMPLE
add_action('manage_posts_custom_column', 'manage_posts_custom_column',10,2);
function manage_posts_custom_column( $column_id,$post_id ) {
	global $typenow;
	switch ("{$typenow}:{$column_id}") {
	case 'listing:business':
		echo '...whatever...';
		break;
	case 'listing:property':
		echo '...whatever...';
		break;
	case 'agent:listing':
		echo '...whatever...';
		break;
	}
}
*/

add_action('restrict_manage_posts','restrict_listings_by_business');
function restrict_listings_by_business() {
	global $typenow;
	global $wp_query;
	if ($typenow=='listing') {
		$taxonomy = 'business';
		$business_taxonomy = get_taxonomy($taxonomy);
		wp_dropdown_categories(array(
			'show_option_all' =>  __("Show All {$business_taxonomy->label}"),
			'taxonomy'        =>  $taxonomy,
			'name'            =>  'business',
			'orderby'         =>  'name',
			'selected'        =>  $wp_query->query['term'],
			'hierarchical'    =>  true,
			'depth'           =>  3,
			'show_count'      =>  true,  // This will give a view
			'hide_empty'      =>  true,   // This will give false positives, i.e. one's not empty related to the other terms. TODO: Fix that
		));
	}
}
add_filter('parse_query','convert_business_id_to_taxonomy_term_in_query');
function convert_business_id_to_taxonomy_term_in_query($query) {
	global $pagenow;
	$qv = &$query->query_vars;
	if ($pagenow=='edit.php' &&
			isset($qv['taxonomy']) && $qv['taxonomy']=='business' &&
			isset($qv['term']) && is_numeric($qv['term'])) {
		$term = get_term_by('id',$qv['term'],'business');
		$qv['term'] = $term->slug;
	}
}
add_action('init','register_listing_post_type');
function register_listing_post_type() {
	register_post_type('listing',array(
		'label' => 'Listings',
		'public' => true,
		'publicly_queryable' => true,
		'show_ui' => true,
		'query_var' => true,
		'rewrite' => true,
		'capability_type' => 'post',
		'hierarchical' => false,
	));
}
add_action('init','register_business_taxonomy');
function register_business_taxonomy() {
	register_taxonomy('business',array('listing'),array(
		'label' => 'Businesses',
		'public'=>true,
		'hierarchical'=>true,
		'show_ui'=>true,
		'query_var'=>true
	));
}

Is this still ok to use this code for current 4.8 Wordpress version?

Thanks chodorowich for your code example in the comments. I noticed that it only filters the first taxonomy if 2 are selected do you have suggestions on how I could fix this?

You should make this into a plugin.

it's much easier now in WP 3.1

function my_restrict_manage_posts() {
    global $typenow;
    $args=array( 'public' => true, '_builtin' => false ); 
    $post_types = get_post_types($args);
    if ( in_array($typenow, $post_types) ) {
    $filters = get_object_taxonomies($typenow);
        foreach ($filters as $tax_slug) {
            $tax_obj = get_taxonomy($tax_slug);
            wp_dropdown_categories(array(
                'show_option_all' => __('Show All '.$tax_obj->label ),
                'taxonomy' => $tax_slug,
                'name' => $tax_obj->name,
                'orderby' => 'term_order',
                'selected' => $_GET[$tax_obj->query_var],
                'hierarchical' => $tax_obj->hierarchical,
                'show_count' => false,
                'hide_empty' => true
            ));
        }
    }
}
function my_convert_restrict($query) {
    global $pagenow;
    global $typenow;
    if ($pagenow=='edit.php') {
        $filters = get_object_taxonomies($typenow);
        foreach ($filters as $tax_slug) {
            $var = &$query->query_vars[$tax_slug];
            if ( isset($var) ) {
                $term = get_term_by('id',$var,$tax_slug);
                $var = $term->slug;
            }
        }
    }
}
add_action('restrict_manage_posts', 'my_restrict_manage_posts' );
add_filter('parse_query','my_convert_restrict');

This code worked perfectly for me in 2020 using WP 5.4! My situation is as follows:
I've created a custom post type Jobs and added two taxonomies (Branche and Status). I needed a filter to show in the admin to quickly filter the status (open/closed) within a branche. Happy to got it working!