and the extant texts of Epicurus do not provide any guidance for this.
This is an interesting perspective because (agreeably so) Epicurus does not prescribe a verbal mantra like the Lord's Prayer, nor strict recommendations about prayer times, or gestures as is the case with Salah in Islam. There might be a lose kind of correspondence between memorizing the Doxai and Christian breviaries, but I might be making a bit of a stretch there.
One important difference is the origin of prayer within the context of each tradition. For Christians, prayer isn't necessarily natural. We only know how to pray (correctly) because a super-being came to Earth, and taught us how. Allegedly, humans existed for thousands of years before any one of them figured out how to pray properly. In this regard, Jesus Christ acts kind of like Prometheus.
With Epicurus, however, prayer is natural, and comes from Nature, and arises spontaneously in human beings the same way we each independently discover masturbation or any other number of development behaviors. In that regard, those philosophers who believe in natural prayer shouldn't prescribe a mantra, because it doesn't come from teachers, its just genetic, human behavior.
He does provide some guidelines, though in terms of how not to get hurt praying: no expectation of wish fulfillment, no expectation of supernatural intervention, no expectation of holding a dialogue.
I think this brings me, personally, to a place where I start to think ... (having been raised as a Methodist) ... what good is prayer if no one is listening? Why bother praying at all if someone isn't considering fulfilling your wishes? Why not just skip the whole, weird, archaic, pseudo-magical ritual altogether? It's not like it's ever been a necessary or important part of my own life.
I entertain the idea that the functional mechanism by which prayer operates is the placebo effect. I think Epicurus whittled down the concept of prayer to the bare-bones: positive thinking is healthy, and believing in a divine nature that you can access through self-improvement is positive. The other stuff (like praying to God for wishes like a genie) are all colorful, cultural additions.
Now, all that being said, let me also add: I don't think most Christians are really reciting their prayer honestly. I think ritualized soliloquies like the Lord's Prayer just puts people's brains on auto-pilot, and they aren't really, devotedly, observantly engaged. So, in that regard, I don't want to give the impression that they are more genuine than our observances. They might be empty.
That's definitely the case with grace before dinner (based my childhood). From personal experience, that's the case with the Pledge of Allegiance in school, too. After a while, they run the risk of losing their authenticity, and become as mundane, barely-conscious, habituated behavior as locking the door when you leave the house. So I think we own prayer just as much as anyone else.