Rule-of-thumb sample size for a z-test

By the usual Central Limit Theorem rule of thumb, how large should a sample be before the sampling distribution of the mean is treated as normal enough to justify a z-test?

$30$. The conventional guideline is $n\ge30$; beyond that the sample mean is close to normal for most populations, and $t$ and $z$ results become nearly identical, converging as $n$ grows.