Student or Learner
Rewrite the sentence:
"The teachers let us have a period off"
Key: We were allowed to have a period off by the teachers
Can I also rewrite it as "We were given a period off by the teachers"?
All the sentences are grammatically correct.
Really neither passive sentence is good English. The point of the passive is to depersonalize an action by removing the subject. "We were given a period off", "we were allowed a period off". In both sentences who exactly authorized the time off is deliberately omitted. If the instrumental phrase "by the teachers" is necessary to convey the full meaning, it is really quite necessary to frame the sentence in the active voice: the teachers allowed/gave us a period off.
Lastly, "allowed" is more formal, "gave" is less formal. Which one is better really depends of the context, regardless of whether it is spoken or written. "Let" in the passive does seem a little stiff.