This belief is based on the idea of God being the ultimate healer, and that people shouldn't intervene with God's workings.
Another more common point of view is that we must not expect God to do everything for us, and that we must take responsibility and help ourselves. This perhaps is the Jewish mentality thus many Jews are doctors. This is an example of emulating God. See the link below for the sentence in the Bible about "walking in His [God's] ways."
I agree with the second point of view. Even with medicine, I believe that God still intervenes and helps with healing, and we should take responsibility to do everything in our power to help ourselves.
If the culture(s) that you are talking about are atheist, I do not see any logical reasons why they believe medicine should not be taken. I doubt that they would do this because it is possible that not taking medicine will strengthen their immune system and in the long run (either through evolution or adaptation) may make them more immune to disease.
Most religious customs that have been commanded (either by God or a religious authority) have good reasons behind them. If this belief is a religious one, then it is most likely a custom rightly or wrongly derived from the Bible or another source and not commanded by God or a religious authority because the phenomenon of not taking medicine as far as I know cannot possibly be derived from any religious document.