The practice of bowing our heads is in a spirit of humility in approaching the throne of God. This was the spirit of the tax-gatherer in Jesus’ parable of the Pharisee and the publican:

LUKE 18:13 "But the tax-gatherer, standing some distance away, was even unwilling to lift up his eyes to heaven, but was beating his breast, saying, 'God, be merciful to me, the sinner!'
LUK 18:14 "I tell you, this man went down to his house justified rather than the other; for everyone who exalts himself shall be humbled, but he who humbles himself shall be exalted."

Praying with head bowed and eyes closed also helps us to focus our minds on our communication with God. Prayer, after all, is “an intelligent exercise of the mind” (Robertson’s Word Pictures).

1 CORINTHIANS 14:15 What is the outcome then? I shall pray with the spirit and I shall pray with the mind also; I shall sing with the spirit and I shall sing with the mind also. [The other side of this verse, singing, is not exclusively between the worshipper and God, but is also a communication from one Christian to another (Eph. 5:19, Col. 3:16).]

I know it is a common trend by “charismatic” groups to display a much more dramatic posture, seemingly in an effort to enhance the experience. Is it necessary to pray with our arms outstretched? After all, Paul said, “Therefore I want the men in every place to pray, lifting up holy hands, without wrath and dissension” (1 Tim. 2:8). The last part of this statement clarifies Paul’s meaning in this instance. A man able to lift up “clean hands” is one who is himself “morally and spiritually clean” (Robertson’s Word Pictures), “without wrath or dissension.” Paul’s instruction is more in regard to a state of mind, then, not a physical posture.

It is important to understand the biblical symbolism of one’s “hand.” “The Lord’s hand” refers to His power (Isa. 59:1, Acts 4:28). “Hands that shed innocent blood” (Prov. 6:17) represent murderers. Likewise, to lift up “holy hands” refers to “the consecrated life, a life devoted to the service of God” (Wayne Jackson, “Should We Lift Up ‘Holy Hands’ When We Pray?,” www.christiancourier.com/questions/holyHandsQuestion.
htm).

Adam Clarke suggests that Paul is making allusion to Jewish tradition.  Under the Old Law, Jews making a sin offering would lay their hands on the head of the sacrificial victim and confess their sins—the animal was then slain as an act of reparation for those sins. “And this very notion is conveyed in the original term epairontav, from airw, ‘to lift up,’ and epi, ‘upon or over.’  This shows us how Christians should pray.  They should come to the altar; set God before their eyes; humble themselves for their sins; bring as a sacrifice the Lamb of God; lay their hands on this sacrifice; and by faith offer it to God in their souls' behalf, expecting salvation through his meritorious death alone” (Adam Clarke’s Commentary).

Simply bowing our heads and closing our eyes conveys the proper attitude and it does not draw attention to one’s self, which is an important thing to consider in a setting of public worship. If I may again cite an observation by Wayne Jackson, “there is no sin in falling upon the ground in prayer (Mark 14:35)”, nor in smiting one’s chest (Luke 18:13), “but would such be wise in a church setting?”