I will expand the answer provided by @DavidKetcheson. First the equations are rewritten as a hyperbolic system of first-order conservation laws:
$$q_t + \nabla \cdot F(q) = 0$$
or
$$q_t + A q_x + B q_y + C q_z = 0$$
Where $q$ is a state vector formed with the components of the stress tensor $(\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{13})$ and components of the velocity vector $(u, v, w)$.
$$q = \begin{pmatrix}
\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{23}\\ \sigma_{13}\\ u\\ v\\ w
\end{pmatrix} \enspace ,$$
$$A = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & {c}_{11} & {c}_{16} & {c}_{15}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{12} & {c}_{26} & {c}_{25}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{13} & {c}_{36} & {c}_{35}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{14} & {c}_{46} & {c}_{45}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{15} & {c}_{56} & {c}_{55}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{16} & {c}_{66} & {c}_{56}\\
\frac{1}{\rho} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & \frac{1}{\rho} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & \frac{1}{\rho} & 0 & 0 & 0
\end{pmatrix} \enspace ,$$
$$B = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & {c}_{16} & {c}_{12} & {c}_{14}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{26} & {c}_{22} & {c}_{24}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{36} & {c}_{23} & {c}_{34}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{46} & {c}_{24} & {c}_{44}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{56} & {c}_{25} & {c}_{45}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{66} & {c}_{26} & {c}_{46}\\
0 & 0 & 0 & \frac{1}{\rho} & 0 & 0 & 0 & 0 & 0\\
0 & \frac{1}{\rho} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & \frac{1}{\rho} & 0 & 0 & 0 & 0
\end{pmatrix} \enspace ,$$
$$C = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & {c}_{15} & {c}_{14} & {c}_{13}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{25} & {c}_{24} & {c}_{23}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{35} & {c}_{34} & {c}_{33}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{45} & {c}_{44} & {c}_{34}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{55} & {c}_{45} & {c}_{35}\\
0 & 0 & 0 & 0 & 0 & 0 & {c}_{56} & {c}_{46} & {c}_{36}\\
0 & 0 & 0 & 0 & 0 & \frac{1}{\rho} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & \frac{1}{\rho} & 0 & 0 & 0 & 0\\
0 & 0 & \frac{1}{\rho} & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix} \enspace .$$
In order to compute the speeds of the problem (as described above) we need to form the matrix $\hat{A}(n_1, n_2, n_3) = n_1 A + n_2 B + n_3 C$, where $\mathbb{n}=(n_1, n_2, n_3)$ is a unit vector and determines the direction of propagation. To find the CFL condition it is necessary to solve
$$\max_{(\theta, \phi)} \max_i \gamma_i(\theta, \phi) $$
where $(\theta, \phi)$ are spherical angles and $\gamma_i$ are the eigenvalues of the matrix $\hat{A}(\theta, \phi)$.
Based on this, and the answer provided by @DavidKetcheson it is simpler to compute the eigenvalues of the Christoffel equation, and solve the optimization problem
$$\max_{(\theta, \phi)} \max_i \lambda_i(\theta, \phi) $$
with $\lambda_i$ eigenvalues of the Christoffel equation. And the speed is just $c = \sqrt{\lambda_i/\rho}$.