Burkert, in typical eloquence, introduces the discussion of prayer, which must accompany pious acts such as libation, sacrifice, and first-fruit offerings (Greek Religion, p. 73):
Any wrong, evil, coarse, or complaining word would be harm, blasphemia, and so the good speech, euphemia, of the participants consists in the first instance in holy silence. Out of the silence then rises up the apostrophe to an Opposite, an invocation and entreaty: the prayer.
Despite the existence of Orphic and Homeric hymns, there was no equivalent to a Book of Common Prayer in ancient Hellas, and none in Hellenismos today). But there were and are common formulas, as well as informal prayer. According to Burkert, the most common ancient formula contained several distinct elements:
Less common in ancient times was the formula of praying for the Good, allowing the god to decide what (if anything) to do. Less formal invocations are also acceptable during the course of normal activity, such as placing the hands to the lips as though blowing a kiss when passing a shrine, and even exclaiming Herakles! in much the same way that many Christians shout Jesus Christ! in a moment of shock.
The posture of prayer is not generally the kneeling posture common among practitioners of some other religions. The Hellenistic prayer posture is more often to stand with arms stretched out. As Harry Thurston Peck points out in Harper's Dictionary of Classical Antiquities:
The Greeks and early Romans stood upright in prayer, extending both the arms, and bringing the hands together with the palms wide open.
Variations on the traditional outstretched arms (a picture of which may be seen at Perseus) are:
Calling out to underworld gods or the dead involves more extreme practices, such as throwing oneself down and pounding fists to the ground. Other unusual but sometimes appropriate practices include silent prayer or writing prayers down in various ways.