There are lots of zero entropy invariant probability measures, many more than just the obvious ones supported on periodic orbits. As you suggest in the question, one can understand the general case by just considering what happens for symbolic systems.
Explicit example: Let $\alpha$ be irrational and let $a_n$ denote the fractional part of $n\alpha$. Consider the sequence in $\Sigma_2 = \{0,1\}^\mathbb{Z}$ given by $x_n = 0$ if $0 \leq a_n < 1/2$, and $x_n=1$ if $1/2\leq a_n<1$. Let $X\subset \Sigma_2$ be the orbit closure of $x=(x_n)$; then there is an entropy-preserving isomorphism between the space of invariant measures for the shift map $\sigma\colon X\to X$ and for the irrational rotation $R_\alpha\colon S^1 \to S^1$. The latter preserves Lebesgue measure on the circle and is uniquely ergodic with zero entropy, so $X$ supports exactly one invariant probability measure $\mu$, which comes from Lebesgue and has zero entropy. Now $\mu$ is a shift-invariant probability measure on $\Sigma_2$ that has zero entropy but is not supported on a periodic orbit.
General result: In fact, the above construction is representative of a general phenomenon. The As RW points out in his answer, you can get lots of zero entropy measures on shift spaces by taking generating partitions for zero entropy transformations. You can even get more, using the Jewett-Krieger embedding theorem (see Petersen's "Ergodic Theory" or Denker, Grillenberger, and Sigmund's "Ergodic Theory on Compact Spaces") Spaces"), which lets you carry out find a closed shift-invariant subset of the shift space that procedure with any zero entropy has the desired measure preserving transformation (among other things)as its only shift-invariant probability measure. So there's a lot there.
