jmc

Since a complex number and its conjugate always have the same magnitude, $$|\overline{9z_1{z}_2+4z_1{z}_3+z_2{z}_3}|=|9{\overline{z}}_1{\overline{z}}_2+4{\overline{z}}_1{\overline{z}}_3+{\overline{z}}_2{\overline{z}}_3|=12.$$From the given information, $z_1{\overline{z}}_1=|z_1{|}^2=1,$ so ${\overline{z}}_1=\frac{1}{z_1}.$ Similarly, $${\overline{z}}_2=\frac{4}{z_2}\text{and}{\overline{z}}_3=\frac{9}{z_3},$$so $$\begin{array}{rl}|9{\overline{z}}_1{\overline{z}}_2+4{\overline{z}}_1{\overline{z}}_3+{\overline{z}}_2{\overline{z}}_3|& =\left|9\cdot \frac{1}{z_1}\cdot \frac{4}{z_2}+4\cdot \frac{1}{z_1}\cdot \frac{9}{z_3}+\frac{4}{z_2}\cdot \frac{9}{z_3}\right|\\ & =\left|\frac{36}{z_1{z}_2}+\frac{36}{z_1{z}_3}+\frac{36}{z_2{z}_3}\right|\\ & =\frac{36}{|z_1{z}_2{z}_3|}|z_1+z_2+z_3|\\ & =\frac{36}{1\cdot 2\cdot 3}|z_1+z_2+z_3|\\ & =6|z_1+z_2+z_3|.\end{array}$$But this quantity is also 12, so $|z_1+z_2+z_3|=\boxed{2}.$