Since the universal set $U={1,2,3,4}$, set $A={1,2,3}$, and set $B={2,3,4}$,

We have $A \cap B = {2,3}$,

Thus, $(A \cap B)^c_U = {1,4}$.

So the answer is: $\boxed{{1,4}}$.

To solve this problem, find the intersection of sets $A$ and $B$, then determine the complement of their intersection with respect to the universal set $U$.

This question tests your understanding of mixed operations involving intersections, unions, and complements of sets. Being proficient in the definitions of each operation is crucial to solving this problem.