3. D.

From the problem, we know that $m, n \in \{1,2, \cdots, 6\}$.
Thus, the total number of events is 36.
The equation has real roots if and only if $m^{2} \geqslant 4 n$, which is equivalent to $n \leqslant \frac{m^{2}}{4}$.
According to this, we can list:
Values of $n$: $1,2,3,4,5,6$;
Number of $m$: $5,4,3,3,2,2$.
Therefore, the total number of favorable events is $5+4+3+3+2+2=19$. Hence, the probability is $\frac{19}{36}$.