The Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909), in Parashat Yitro (Halacha 1), discusses the laws concerning the direction one must face while praying the Amida (listen to audio recording for precise citation). He notes that the prayers we recite take the place of the sacrifices that were once offered in the Bet Ha’mikdash, and our lips are the "altar" upon which the "sacrifice" is offered. In light of this direct relationship between prayers and sacrifices, one must recite the Amida while facing toward the direction of the Bet Ha’mikdash, the site where the sacrifices were offered. The Ben Ish Hai also adds that the site of the Bet Ha’mikdash is the place from where all prayers ascend to the heavens. For this reason, the Temple Mount is referred to as "Talpiyot," a combination of the words, "Tel Piyot" – the "mountain of mouths" – referring to the fact that all mouths are directed to that site in prayer.
Therefore, one who lives outside Israel should pray in the direction of Israel; in New York, for example, one faces east. During a flight to Eretz Yisrael, one prays facing the front of the plane, while during a flight away from Eretz Yisrael, one faces the back of the plane. One who prays in Israel should face Jerusalem. Thus, for example, a person in the Northern Israeli city of Tiberias should face south, while somebody praying in the southern city of Be’er Sheba faces northward. In Jerusalem, a person prays toward the Temple Mount, where the Temple stood.
The Ben Ish Hai adds that besides positioning oneself toward Jerusalem, one should also have in mind while praying that he prays toward the site of the Kodesh Ha’kodashim (the innermost sanctum of the Mikdash). He therefore advises writing on a slip of paper the words, "Eretz Yisrael, Jerusalem, Bet Ha’mikdash, Kodesh Ha’kodashim" and keeping that slip of paper in one’s Siddur. This helps ensure that at least when one begins the Amida he will think in his mind that he prays toward the sacred site of the Kodesh Ha’kodashim. Many of our Siddurim today (including the Abodat Hashem edition of the Siddur) indeed print these words on the pages of the Amida, or at the beginning of the Amida.
Some synagogues, due to constructional restrictions, are unable to position the Aron Kodesh (ark) in the direction of Jerusalem. Therefore, one should not necessarily assume when he enters a synagogue that he should face toward the Aron Kodesh while he prays. A person who visits a synagogue should ensure to face the proper direction, rather than assume that by facing the ark he faces toward Israel.
Summary: One must pray the Amida facing toward the direction of Israel; within Israel, one faces toward Jerusalem, and in Jerusalem one faces toward the site of the Temple. One should also have in mind during the Amida that he prays toward the site of the Temple, and for this reason many Siddurim print the words "Israel, Jerusalem, Bet Ha’mikdash" near the Amida to remind a person to have this intention.
