Tehillim 137 is printed in several bentchers before birkas hamazon, but from what I have seen most people do not say it. What is the correct thing to do: say or omit? Are there different minhagim?
In my Sefer Ish Emunot (Orah Haim 1:25) I wrote a pretty lengthy discussion about this. First of all, we have to know the Mishna Berura (Orah Haim 1:11) brings the Shela (Shaar HaOtiot 85a) that brings down to say Al Naharot on days when Tahanun is said. However, on days where it is omitted, he says to say Beshuv Hashem etc (sources to support the M"B - Kaf HaHaim Sofer 157:18. More BaEsba 3:113 writes to say the words "Im Eshkachech", and Rav Yaakov Hilel Shelit"a in his Sefer Amude Horaah 3:44 brings a source from the Arizal in Shaar HaMiswot 45a. Kaf HaHaim Palagi 36:73. It is also Mashma from the Magen Avraham 292:5 that there is a Minhag to say it).
[Also, the Zohar (Teruma 257b) writes that when one is eating a meal, at some point, he must mourn the Bet HaMikdash.]
However, HaRav Musafi Shelit"a (Shivat Sion 1:6:40) writes that it is only a Minhag Ashkenaz (so wrote the Ben Ish Hai Wayelech #4).
However, a Sefaradi that follows this Pesak should not refrain from saying Lamnaseah and Avarecha (see Siporen Shamir 6:68 and Shivat Sion 1:6:39).
Summary: the Minhag of Ashkenaz is to say it on a day when Tahanun is recited. While, the mainstream Sefaradi approach is "Shev WeAl Taase Adif" and in conclusion Sefaradim do not recite .
[The Peri Megadim (1:5) holds that on weekdays Al Naharot Bavel should be recited, and on Shabbat Shir HaMaalot should be recited.]
The Chabad custom is to say it on days when Tachnun is recited.
In his footnotes to the Siddur, Dayan Raskin says that this practice comes from the Zohar (II:157B) that says that when someone finds pleasure in his food he should remember and worry about the holiness of the Land of Israel and and the Palace of the King that was destroyed.
The Arizal and the Shaloh use this reason to explain why we say Psalm 137 during the weekday and Psalm 126 on Shabbat.
The Seder Hayom finds a hint to this practice (of saying Psalm 137) from the fact that the end of Psalm 136 (verse 25) talks about G-d giving us food:
136:25 Who giveth food to all flesh, for His mercy endureth for ever.
136:26 O give thanks unto the God of heaven, for His mercy endureth for ever.
137:1 By the rivers of Babylon, there we sat down, yea, we wept, when we remembered Zion.