How did pilots avoid thunderstorms and related weather before “reliable” airborne weather radar was introduced on airliners?
They looked out the window! 

If you are talking about a developing thunderstorm, especially during the day, they can be quite easy to see in front of you provided decent visibility. Here is some good footage of what it looks like to fly around a towering cell, you will note its fairly easy to see. At night, lightning, even in the clouds tends to be a prominent effect and is a good indicator of "dont fly there". Here is a good picture of that effect (as well as a little story about flying around it). 

Of course, summer haze, the dark of a moonless night, shifty weather, or just plain old mistakes sometimes lands planes right where they should not be sometimes with disastrous results.

Here is an interesting report on thunderstorms and aircrafts from 1948 that may have some interesting info for you.