Don't upvote this. I just figured since this is getting some attention from another question on MO I'd communicate the blurb that is going to go into what I'm writing now. (See also a related bit on meta.) A comment won't allow TeX at this point, hence the use of an answer.

Given a Markov partition $\mathcal{R} =$ {$R_1,\dots, R_n$} for the cat map and $m \ge 3$, consider the sets $R_{jk} := R_j \times$ {$\frac{k}{m} - j\epsilon$}, where $1 \le k \le m$ and $\epsilon < \frac{1}{mn}$. The family $\mathcal{R}' :=$ {$R_{jk}$} is readily seen to be a proper family for the cat flow. [A. Gogolev, private communication] The Poincaré map for $\mathcal{R}'$ sends $R_{jk}$ to $R_{j,k+1}$ for $1 \le k \le m-1$, and because $\mathcal{R}$ is a Markov partition for the cat map it follows that $\mathcal{R}'$ is a Markov family for the cat flow.