Solving multiple instances of 3SUM generated from the same set
It's not linear time, but here at least is an $O(n^{3/2})$ randomized solution to the problem in which you are given a set of $n$ elements together with the sums of $n$ randomly-generated triples of the elements, and you have to find a triple that matches one of the given sums.


Pick randomly two subsets $A$ of $n^{3/4}$ elements and $B$ of $n^{1/2}$ elements. With constant probability at least one of your 3SUMs is in $A+A+B$.
Compute the set $A+A$ and store it in a hash table for fast lookups.
For each pair of a value $x$ in $B$ and a target value $z$ of one of the input 3SUMs, test whether $z-x$ is one of the stored values of $A+A$, and if so whether the resulting triple(s) of numbers $A+A+B$ that sum to $z$ are distinct from each other.


I originally thought this would work equally well for an adversarially-selected set of 3SUMs (instead of the randomly-selected set in your question), but now I'm not sure sure about that part: the expected number of 3SUMs you hit with your choice of $A+A+B$ is always constant, but maybe the adversary could fix things so that you have a high probability of hitting none of them and a small probability of hitting many at once? I'm not sure this is possible but it would take more analysis than I've done so far to be certain.