1. Represent the consecutive integers as $x$, $x+1$, and $x+2$.
2. Write the equation for their sum:
x + (x+1) + (x+2) = 84
3. Simplify and solve for $x$:
3x + 3 = 84
3x = 81
x = 27
4. The integers are $x = 27$, $x + 1 = 28$, $x + 2 = 29$.
5. Therefore, the three consecutive integers are 27, 28, and 29.