The equation is equivalent to $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{4}{3}$ This is a standard equation in the form of $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n}$$ Its solution is discussed in the book  Indeterminate Equation .

Without loss of generality, let's assume $x\le y\le z$, then $\frac{1}{x}\ge\frac{1}{y}\ge\frac{1}{z}$, therefore $$\frac{1}{z}\ge\frac{1}{3}\cdot\frac{4}{3}=\frac{4}{9}\implies x = 1, 2$$

$\underline{When\ x=1}$ We have $\frac{1}{y}+\frac{1}{z}=\frac{1}{3}$. Therefore, $$\frac{1}{y}\ge\frac{1}{2}\cdot\frac{1}{3}=\frac{1}{6}\implies y = 1, 2, 3, 4, 5, 6$$ When $y=1,2,3,5$, there is no positive integer $z$ satisfying the given relation. When $y=4$ and $6$, $z$ equals $12$ and $6$ respectively.

$\underline{When\ x=2}$ We have $\frac{1}{y}+\frac{1}{z}=\frac{5}{6}$, Therefore, also noting the assumption of $y\ge x$, $$\frac{1}{y}\ge\frac{1}{2}\cdot\frac{5}{6}=\frac{5}{12}\implies y=2$$ When $y=2$, we have $z=3$. Hence, all the solutions are $(1, 4, 12), (1, 6, 6), (2, 2, 3)$ and all their permutations.