In between the 'Shir HaMaalot' that precedes birchat hamazon and the the blessings themselves, there are a few psukim beginning תהלת ה' ידבר פי . Some people add these, and some don't. What's the origin of the custom? Which groups practice it?
According to Rabbi Ari Enkin (link) "The origin for reciting these verses is unclear." He says that the Kaf Hachaim O.C. 157:22 notes this minhag. However, the Kaf Hachaim there (link) lists a number of pesukim to say, two of which are part of the four one is likely to hear nowadays:
I'm not familiar with the sources that the Kaf Hachaim quotes here.
The question of Tehillath HaShem is actually what sparked my question yesterday about R' Hirsch. I was told that Shir HaMa'aloth was the favorite hymn of the early (proto?-)Zionists, and so R' Hirsch, the anti-Zionist that he (apparently) was, insisted on adding "legitimate" words of praise to G-d prior to Birkath HaMazon, in order to separate it from the paragraph usurped by the unsavory Zionists.
I have no source, as I would like more information myself. I made this an answer instead of a comment for two reasons.