Solution:

Factor the given least common multiples: $$72=2^3\cdot3^2,\quad 600=2^3\cdot3\cdot5^2,\quad 900=2^2\cdot3^2\cdot5^2.$$ Since $\operatorname{lcm}(x,y)$ has no factor of $5$, neither $x$ nor $y$ has a factor of $5$, and $z$ must contain $5^2$.

Write $x=2^a3^b$, $y=2^c3^d$, and $z=2^e3^f5^2$. The power of $3$ in $\operatorname{lcm}(y,z)$ forces $d=2$, and the power of $2$ in $\operatorname{lcm}(x,z)$ forces $a=3$.

The remaining independent conditions are $\max(b,f)=1$ and $\max(c,e)=2$. These have $3$ and $5$ ordered choices, respectively, so there are $3\cdot5=15$ triples.

Thus, the correct answer is A.