The headlines are a bit over the top:
Religious brains more calm in face of anxiety: study
Religion protects against stress, study suggests
Believers stay calmer than atheists in trying situations, study suggests
That last one comes closest to the truth, in terms of headlines. From the conclusion of the actual study, which you can find here.
Our results indicate that religious conviction is associated with reduced neural responsivity to uncertainty and error on a generic decision-making task.
So, yes, religious people are less anxious. I don't know how far to take the significance of the results when the sample size is 28 people and only four are "non-religious", but I'll point this much out anyway: the anxiety they're measuring is the anxiety experienced after making a mistake.
Yes, you read that right. The primary and most significant conclusion of the study is that atheists and other non-believers experience a sharper pang of anxiety or stress in the milliseconds following an error.
Frankly, I find that disturbing but not surprising. It means that religious people make mistakes and don't seem to care about it as much. If you've ever studied any history, you certainly won't find it hard to believe that religious people gloss over their mistakes more easily than others.
Maybe it just takes practice. After all, if you can read all four Christian gospels with their accounts of the death and resurrection of Jesus Christ without feeling anxiety about all of the direct contradictions, I suppose you just build up an immunity to facts or something.
I'll take my anxiety over their certainty any day of the week - even Sunday
Update: A more thorough analysis The Neurocritic: Atheists Are Neurotic and Religious Zealots Are Antisocial