From The Seven Colors ot the Rainbow: Torah Ethics for Non-Jews by Rabbi Yirmeyahu Bindman (also contains Kabbalah/mystical references for those who are interested)
"...Non-Jews are not obligated to make a quorum or to appoint times, but if they prefer the atmosphere of communal prayer then they may make an assembly at any convenient time or place, as groups who uphold the Seven Laws customarily do. Special prayers have been composed according to the Noachide concepts to help them in this, but they are under no obligation to use them. They must only be prepared to pray, without an intermediary, to G-d Himself alone, as the Creator of all and the only Being who hears prayers and supplies the wants of humanity." (p 101)
From Path of the Righteous Gentile: An Introduction to the Seven Laws of the Children of Noah by Chaim Clorfene and Yakov Rogalsky Chapter 15 (some of the book can be found online at gopher://gopher.chabad.org/11/outlook/7laws but not all...this part is not on-line)
They attribute this chapter to Rabbi Moshe Feinstein of blessed memory, and have translated freely from his responsa on this matter.
"The Noahite's prayer should not consist soley of supplications but should also include praises to G-d [BTW, the same holds true for Jewish prayer]
"The act and experience of praying to G-d (and it should be obvious that it is forbidden to pray to any being other than G-d) has limitless levels. Whether one supplicates G-d for his needs and wants, or for help in times of danger or stress, or engages in deep meditational prayer in order to elevate oneself spiritually, prayer is always a mystical expericnec, a communion with the infinte Creator of one's own soul....
"And King David wrote, 'Praise the L-RD, all nations, extol Him all peoples' (Ps. 117:1) This verse from Psalms refers specifically to the prayers of the Children of Noah." (p 126)
I would suggest that a Noachide select Psalms that praise G-d in a general way, prayers from the Jewish Siddur minus Jewish specific things. A Noachide can also make their own prayers giving thanks to G-d and asking for His protection and blessing. As in all things, consult your local rabbi.
