The more idiomatic translation of "de rien" would be "think nothing of it".
Source: lifelong bilingual in French & English.
Source: lifelong bilingual in French & English.
As cced already said, common ways of expressing this in American English include “don’t worry about it”, “no worries”, “no big deal”, “no problem”, “you’re welcome”, “my pleasure”, etc.
French also has more built-in structures for formal address/politeness, though many of them are being left by the wayside just as we left feudal lordships with a peasant class.
For what it's worth, I'd say that the more "stuffy" version of "de rien" in French would be "il n'y a pas de quoi".