Every couple you will photograph is unique. The challenge is that "unique" doesn't always translate to "natural and relaxed posing". Some couples will struggle a bit (or a lot) with relaxing and posing naturally in front of the camera.
Is There A Key?
Unfortunately, there is no one-size-fits-all-easy-and-simple solution. It would be nice if the solution was simply to have a squeaky toy ready - but, if that was the case, all the photographers would already be doing it.
Here are some of the little techniques I've learned over the years.
Have Proper Expectations
Realizing that some couples will be easier to work with and some will be harder can help to have a proper mindset. At the same time, you are the "professional" and the couple isn't. You're job is to help them be the best they can be without stress. You'll have to work harder with some couples, but, that's just the way it is.
Make a Good First Impression
This is a bit basic, but greet the couple with a friendly smile and warm handshakes. You should be genuinely happy to see them! People will subconsciously pick up on nervousness. If you are nervous they will be more nervous and tense during the photo session.
Social Skills 101
One more REALLY basic suggestion (which, you are probably already doing): carry on a pleasant conversation with them throughout the entire photo session. If the only time you talk to them is when you give them an instruction "Lean towards each other, rotate to the side a bit more, more, no, too far!" you'll end up with an uptight couple.
Most couples love to talk, and their favorite subject will likely be themselves. To be honest, I have never had to try and be interested in the couples I've worked with. I have genuinely enjoyed getting to know each and every couple. I've loved finding out their story: when and how they met, how long they've been engaged, how they became engaged, what day the wedding is planned for, etc.
Start With Warm Ups
I don't mention this to the couple, but I generally plan the first part of the photo session with the simple goal of relaxing the couple. I don't try for my best shots during those first 15~20 minutes. My goal is simply to put them at ease with me and the camera. I know that if they can become relaxed during the first 15 minutes, the remaining hour or two of photos will go much more smoothly!
During that first segment I try to use basic and simple poses. I spend more time chatting and less time snapping photos. I encourage them in and reinforce their posing ideas.
Unexpected Photos
Another way to refer to these would be "outtakes". It's a fun method to try and it's success will depend on the couple and how they interact with each other when they aren't expecting you to be taking their pictures. I have found that some couples will talk between themselves while aimlessly starring off into space, while others will have an animated and happy conversation while looking into each other's eyes and occasionally laughing (better chance of success with the latter).My goal: to take several photos while they are fully relaxed and are not expecting me to be taking their picture.
The method: I usually try this out while I'm changing memory cards or rummaging for something in my camera bag. I'll let them know that I need to get something out of my camera bag or do something with my lens and that they can relax for a minute. I rummage around in my bag while looking for the new card; and while I am doing so I watch the couple out of the corner of my eye.
The challenge: taking a few photos can draw their attention back to the camera. Sometimes I have a wireless photo release already on the camera and will use that (especially if the camera is on a tripod). Other times I have an assistant with me and will have them take a few of these shots while I'm rummaging in my camera bag. Sometimes the couple so completely forgets about the camera and me that I can simply go back to the camera and press the shutter release without them noticing.
Hopefully, these suggestions, when combined with your personality and experience, will help you take even better engagement photos of couples that appear relaxed and natural!