One neat way of sampling out of the Cauchy (and, actually, student as well) multivariate distribution exploits its (their) ellipticity. In order to sample an elliptical random vector it is enough to proceed along the following steps:

• generate a point $x$ uniformly on the unit sphere of $\mathbb{R}^{n}$; with $n$ being the dimmension of your r.v.
• independently on $x$, generate a real number $\alpha$ according to some univariate law (in your case, it is Cauchy with $\gamma=1$)
• compute $\alpha x$ and transform the resulting point according to the matrix $\Sigma$ (via a matrix $A$ such that $AA^{T}=\Sigma$)
• advance the resulting point by your desired mean $\mu$

Not really an answer. One neat (Edit: not so neat in this case) way of sampling out of the Cauchy (and, actually, student as well) multivariate distribution exploits its (their) ellipticity. In order to sample an elliptical random vector it is enough to proceed along the following steps:

• generate a point $x$ uniformly on the unit sphere of $\mathbb{R}^{n}$; with $n$ being the dimmension of your r.v.
• independently on $x$, generate a real number $\alpha$ according to some univariate law (in your case, it is Cauchy with $\gamma=1$ Edit: as fedja pointed out, this is not so simple -- actually, one has to sample from absolute value of $n$-variate Cauchy, which seems to be the main obstacle here.)
• compute $\alpha x$ and transform the resulting point according to the matrix $\Sigma$ (via a matrix $A$ such that $AA^{T}=\Sigma$)
• advance the resulting point by your desired mean $\mu$