Learn Spanish (Latin America): Nosotros no temíamos a los profesores estrictos.

The imperfect tense is used for ongoing, habitual, or background states in the past. Here, fearing the teachers wasn't a one-time event that started and ended; it was a general attitude or ongoing state of mind over a period of time.

Why does temíamos have an accent mark on the letter 'i'?

For regular -er and -ir verbs in the imperfect tense, every single form has an accent mark on the í in its ending (like -ía, -ías, -íamos). This ensures you put the spoken stress on that letter.

What is the word a doing in the phrase temíamos a los profesores?

This is the personal a. In Spanish, when the direct object of a verb is a specific person (or a group of people, like los profesores), you must put the word a right before it. It does not translate to anything in English.

Would I still use a if they feared something that wasn't a person?

No, you wouldn't. The verb temer (to fear) does not automatically require a after it. If you were talking about fearing a thing, like a storm, you would just say temíamos la tormenta without the personal a.