It is customary recite a preliminary chapter of Tehillim before reciting the birkat hamazon after meals. On weekdays, the general custom is to recite chapter 137, “Al Naharot Bavel”, which is intended to remind us of the destruction of Jerusalem and the current exile. The Zohar states that one who derives pleasure from bread and enjoys the taste of foods is required to remember the Holy Land and the Beit Hamikdash after every meal. On Shabbat, and other festive days when such sobering thoughts are not appropriate, chapter 126, “Shir Hama’alot”, is recited instead which speaks about the future redemption. Some also recite “Shir Hama’alot” when in the presence of a special guest. These Tehillim should be recited before washing one’s hands at the end of the meal (“mayim achronim”).
Click here to read moreIt is recorded that there were great rabbis who had the practice to recite “Shir Hama’alot” every single day of the year before reciting birkat hamazon.  The only exception to this was on Erev Tisha B’av, when they too would recite the “Al Naharot Bavel” in order to better prepare themselves emotionally for the onset of Tisha B’av. It seems that that this was also the custom of the Baal Shem Tov, the Arizal, and other kabbalists. The reason for the custom to always substitute “Al Naharot Bavel” with “Shir Hama’alot” was in order to be “b’simcha”, to be happy whenever possible, and reciting “Al Naharot Bavel” can’t help but make one sad. Many Sefardim and Chassidim also recite “Lamnatze’ach” and other preliminary readings before the birkat hamazon. Reciting a chapter of Tehillim before the birkat hamazon also allows one to fulfill the requirement that every meal include words of Torah.
Based on the teachings of the Arizal, some individuals also have the custom of reciting additional Scriptural verses prior to reciting the birkat hamazon, most notably: “Tehillat Hashem”, “Hodu L’hashem”, and “Mi Yemalel”, among others. The origin for reciting these verses is unclear. There is reason to suggest, however, that these verses were regarded as a segula, that one should always have food to eat. As such, it is especially appropriate to recite them during one’s meal or as part of the birkat hamazon in the hopes that one will never go hungry. It is also suggested that these supplementary verses were chosen because they parallel the themes of the blessings of the birkat hamazon. Those who recite these additional verses should do so after having washed mayim achromin. One should also take a few moments before birkat hamazon to contemplate the great mitzva which one is about to perform. 
*********************************************** Mishna Berura 1:11
 Rite and Reason p.193
 Magen Avraham O.C. 1:5, Aruch Hashulchan O.C. 181:9
 Rite and Reasom p.199 note 18
 Aruch Hashulchan 181:9
 Piskei Teshuvot 182:note 60
 Minhag Yisrael Torah 1:3
 Kaf Hachaim O.C. 157:22 (they are from Tehillim 145:21, 115:18, 107:1, 106:2)
 Otzar Hatefillot siddur (Ashkenaz, vol. 1 p. 474), cited by Rabbi Gil Student at: http://hirhurim.blogspot.com/2005/09/tehillas-hashem.html
 Rite and Reason p.140
 Kaf Hachaim O.C. 157:22
 Kaf Hachaim O.C. 183:46