Cover Time of Directed Graphs
Clearly polynomial mixing time implies polynomial cover time. (Well, not in general. We need the stationary probability at least $1/poly(n)$ at each vertex.) So check Mihail's paper Conductance and convergence of Markov chains-a combinatorial treatment of expanders which proves rapid mixing of regular directed graphs and general Markov chains based on conductance.

Also see the paper Pseudorandom walks on regular digraphs and the RL vs. L problem by Reingold, Trevisan, and Vadhan. Following Mihail's work. They defined the parameter $\lambda_\pi(G)$ which is equivalent to $\lambda_2(G)$, the second largest eigenvalue in absolute value, when the graph $G$ is time-reversible, and remains well-defined for general Markov chains. This parameter is then used to bound the mixing time of $G$.