Solution:

Recall the four following rules:

• odd plus odd and even plus even is even

• even plus odd is odd

• even times anything is even

• odd times odd is odd

These rules can be easily verified by representing arbitrary odd numbers as $2m+1$ and arbitrary even numbers as $2n$ respectively, for integers $m,n.$

With this in mind, let's examine each answer choice individually:

A:

Note that $3$ is odd. This gives us $$\text{odd} + \text{odd} \cdot \text{odd}.$$ From our above rules, we know that this is even.

B: $$\text{odd} \cdot \text{odd} - \text{odd}.$$ Once again, this is even.

C: $$\text{odd} \cdot \text{odd}^2 + \text{odd} \cdot \text{odd}^2.$$ This is also even.

D: $$(\text{odd} \cdot \text{odd} + \text{odd})^2.$$ Unfortunately, this is also even.

E: $$\text{odd} \cdot \text{odd} \cdot \text{odd}.$$ This is odd.

Therefore, E is the only answer choice that is an odd integer.

Thus, E is the correct answer.