If you set $\frac{x}{yz} = p$, $\frac{1}{y} = q$, $\frac{1}{z} = r$, then your equations are
$p + q + r = k_1$
$p^2 + q^2 + r^2 = k_2$
$p^3 + q^3 + r^3 = k_3$
Using $(p+q+r)^2 - (p^2 + q^2 + r^2) = 2(pq + qr + rp)$ we get
$pq + qr + rp = (k_1^2 - k_2)/2$
Similarly we can find the value of $pqr$.
Thus we can find the polynomial $(x-p)(x-q)(x-r)$, by finding its coefficients. Once we have the polynomial, since it is a cubic, it is solvable by standard methods.
Once we find $p,q,r$, finding $x,y,z$ should not be difficult (keep in mind the permutations possible).
See: Newton's identities for a way to find the coefficients.
See: Roots of cubic for a formula for the roots of cubic.