1)The Shulchan Aruch writes that it is proper for all those who are yorei shomayim (possess a fear of heaven) to feel pain and worry over the destruction of the Bais Hamikdash. One of the ways that this was accomplished was by saying Tikkun Chatzos although practically this is difficult as it needs to be recited at midnight.
2) One of the other ways of accomplishing this, is the recital of Al Naharos Bavel before Birchas Hamazon. The Mishan Berura rules that this is not said on Shabbos or Yom Tov or any day that tachanun is
not recited and is replaced with Shir Hama’alos instead.