Is there a general criterion for a continuous map on a compact metric space to have every ergodic measure purely atomic? For example, a circle rotation with rational rotation number has this property (each ergodic measure is an atomic measure supported on a periodic orbit).

The atomic invariant measures are exactly the ones supported by periodic orbits, so your condition is equivalent to $\mu(X\setminus P)=0$ for all ergodic $\mu$. Here $P$ denotes the collection of periodic points.

This happens if and only if the orbit of every $x\in X$ spends almost all its time near $P$. More precisely, it happens if and only if $$ \frac{1}{N} \# \{ n: 1\le n\le N, T^n x\in K \} \to 0 \quad (N\to\infty) $$ for all $x\in X$ and all compact $K\subseteq X\setminus P$. (If you have an $x, K$ violating this condition, take a limit point of $(1/N)\sum_{n=1}^N \delta_{T^n x}$ to obtain an invariant measure $\mu$ with $\mu(K)>0$. For the converse, use the ergodic theorem.)