I have a MedicAlert® bracelet. Firstly, it's a convenient item that first-aiders can check if you are unable to speak for yourself, which describes conditions you may have. It also contains a phone number for more information which they could use.
There are downsides - it requires you to wear it the whole time. I don't even notice mine any more, but I can't imagine having to wear one around my neck - it'd get frustrating, surely. In addition, items around the neck are more often perceived as jewellery, and in theory uneducated persons might consider it as potentially worth stealing. However it's just stainless steel, and few people have ever asked what it is - more people have seen the logo, recognised and asked what it's for.
I quickly looked and they now come in a variety of shapes and styles for men - far more than when I first got mine.
They started in California, USA and now have nine international affiliates around the world.
Is it important to wear while travelling? I'd say it depends what conditions you have. Ones which might have you unconscious or require urgent care, or where you have strong allergies to medicines you might not realise they're giving you, for example, would be a yes. These are lifelong conditions. Certainly if you're also taking medicine that you require daily, it's useful to have as well, so that if you're unconscious and people don't realise what you're missing, they can start to act on it.