Having a "responsive" site is not enough. I have many clients with sites that are responsive but are not passing the test parameters.
The first thing you need to do is to find out what Google thinks of your site in terms of mobile-friendliness right now. There are two parts to this. You can test your home page (or any page) using this Google Mobile Friendly Test site:
You also need to hook your site up through Google Webmaster Tools - GWT will give you a breakdown for all pages on the site, not just your home page. It is important that you get a clean bill of health across all the pages on the site, not just the Home Page.
For WordPress sites that are failing badly, you can implement WP Touch and create a mobile friendly site which will still utilize the same URL and content from the main site. Otherwise, your main site can usually be "tweaked" to pass the test.
Non-WP formats are much trickier. In many cases, it is going to require a site rebuild. You can utilize an m.mysite.com mobile site but this becomes cumbersome as you have to update 2 sites now every time you want to add/change content.
Hope this is helpful.