Let's denote the three consecutive odd integers as \( x \), \( x + 2 \), and \( x + 4 \).
We know that their sum is given by:
\[ x + (x + 2) + (x + 4) = -21 \]
Now, simplify the left side:
\[ 3x + 6 = -21 \]
Next, we will isolate \( x \) by first subtracting 6 from both sides:
\[ 3x = -21 - 6 \]
\[ 3x = -27 \]
Now, divide both sides by 3:
\[ x = -9 \]
Now we can find the three consecutive odd integers:
The first integer is \( x = -9 \).
The second integer is \( x + 2 = -9 + 2 = -7 \).
The third integer is \( x + 4 = -9 + 4 = -5 \).
Thus, the three consecutive odd integers are:
\[ \boxed{-9}, \quad \boxed{-7}, \quad \text{and} \quad \boxed{-5} \]
To verify, we check their sum:
\[ -9 + (-7) + (-5) = -9 - 7 - 5 = -21 \]
The sum is indeed \(-21\), confirming the integers are correct.