To use the method of contradiction, we must assume the opposite of what we want to prove and show that this assumption leads to a contradiction.

We want to prove that "If $a, b, c < 3$, then at least one of $a, b, c$ is less than $1$." The opposite of this statement is "If $a, b, c < 3$, then $a, b, c$ are all not less than $1$," because it denies the existence of any number among $a, b, c$ that is less than $1$, while still adhering to the condition $a, b, c < 3$.

So, to use the method of contradiction, we assume that all of $a, b, c$ are not less than $1$, which could mean they are greater than or equal to $1$. Now we need to find a contradiction under this assumption.

Assuming $a, b, c \geq 1$ and $a, b, c < 3$, we consider the sum of $a, b$, and $c$. Their minimum possible sum occurs when $a=b=c=1$, which is $1+1+1=3$, and their maximum possible sum is less than $3+3+3=9$ since each of $a, b, c$ is less than $3$.

However, if $a, b, c$ are all greater than or equal to $1$, their sum must be greater than or equal to $3$, which violates the original condition that they must all be less than $3$. This contradiction implies that our initial assumption is false. Therefore, it is indeed true that if $a, b, c < 3$, at least one of $a, b, c$ must be less than $1$. The correct choice is:

$$\boxed{D: Assume \ a, b, c \ are \ all \ not \ less \ than \ 1}$$