How do you update user_email on the front end in WP 3.3?

You need to use wp_update_user() for the email, as it is not user-meta but core user data. The code should look something like this:

$args = array(
    'ID'         => $current_user->id,
    'user_email' => esc_attr( $_POST['user_email'] )
);
wp_update_user( $args );

Note: that’s untested, but it should work out of the box.