What is an efficient way to query based on post_meta?

The problem I am having is when I allow users to filter posts it overpowers the database and things run extremely slow. I am unsure what the best way to query posts using meta values is. My site has a form that users can submit to filter posts based on checkboxes containing meta values. I currently have the site making an array out of the checkboxes selected for individual post_meta keys like so:

for ($i=0; $i< count($amenities); $i++)
    {
        $count = count($arrays);
        $arrays[$count] = array(
                    'key' => 'amenities',
                    'value' => $amenities[$i],
                    'compare' => 'LIKE'
                    );
    }

Then I query the database using an array of arrays which contain the values selected for each meta_key. Like this:

$the_query = new WP_Query(array(
        'post_type' => 'listing',
        'post_status' => 'publish',
        'category_name' => 'private_rental',
        'orderby' => 'rand',
        'meta_query' => $arrays
    ));

When I do this I get the results I want but it can take over a minute for the results to appear on the screen. Is there a more efficient way of doing this?

Any help would be appreciated!