We use the inequality of arithmetic and geometric means for four positive terms:
$$\frac{a_1 + a_2 + a_3 + a_4}{4} \ge \sqrt[4]{a_1 a_2 a_3 a_4},$$
where equality holds when $a_1 = a_2 = a_3 = a_4$.

Applying this to each factor:
$$(3x + y) \ge 4\sqrt[4]{x^3 y}$$
$$(3y + z) \ge 4\sqrt[4]{y^3 z}$$
$$(3z + x) \ge 4\sqrt[4]{z^3 x}$$

Multiplying these inequalities:
$$(3x + y)(3y + z)(3z + x) \ge 4\sqrt[4]{x^3 y} \cdot 4\sqrt[4]{y^3 z} \cdot 4\sqrt[4]{z^3 x} = 64\sqrt[4]{x^4 y^4 z^4} = 64xyz.$$

Equality holds when $x = y = z$.