Remove classes from body_class

You can configure the $whitelist array in this function to filter out all other unwanted classes.

add_filter( 'body_class', 'wpse15850_body_class', 10, 2 );

function wpse15850_body_class( $wp_classes, $extra_classes ) {

    // List of the only WP generated classes allowed
    $whitelist = array( 'portfolio', 'home', 'error404' );

    // Filter the body classes
    $wp_classes = array_intersect( $wp_classes, $whitelist );

    // Add the extra classes back untouched
    return array_merge( $wp_classes, (array) $extra_classes );
}