Example 3 Find the positive integer solutions of the equation $\frac{x y}{z}+\frac{x z}{y}+\frac{y z}{x}=3$.

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Solution: The original equation can be transformed into $x^{2} y^{2}+x^{2} z^{2}+y^{2} z^{2}=3 x y z$. By the AM-GM inequality, we have $x^{2} y^{2}+y^{2} z^{2} \geqslant 2 x y^{2} z$,
$$
y^{2} z^{2}+z^{2} x^{2} \geqslant 2 x y z^{2}, x^{2} y^{2}+z^{2} x^{2} \geqslant 2 x^{2} y z .
$$

Thus, $x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2} \geqslant x^{2} y z+x y^{2} z+x y z^{2}$,
which means $3 x y z \geqslant x^{2} y z+x y^{2} z+x y z^{2}$, leading to $3 \geqslant x+y+z$. Since $x, y, z$ are all positive integers, then $x=1, y=1, z=1$.
Therefore, the positive integer solution to the original equation is $(1,1,1)$.