My feeling is that it may be premature to declare what types of problems look tractable with respect to the polynomial method. For instance, the idea of using the polynomial method to attack the finite field Kakeya problem, while "obvious" in retrospect, was certainly a big shock to many of us working on the Kakeya problem at the time Dvir's argument came out.
In contrast, the question of getting a non-trivial Szemeredi-Trotter or sum-product theorem in finite fields, while closely related to the Kakeya problem (see e.g. my paper with Bourgain and Katz on this topic), has so far resisted all attempts at a polynomial method proof. But this could simply be because we haven't yet found the right way to generate the right sort of polynomials for this problem. Similarly, the capset problem of determining better bounds on $r_3(F_3^n)$ than what one can get from Fourier methods is one that at first looks very amenable to a polynomial method approach, but again there has been no progress on this front. (These are great problems to look at, by the way, if someone in this area is looking for a high-risk, high-reward task to add to their research projects.)
What I would like to see more of in the future is more development of the somewhat vague idea of the "Zariski complexity" of various sets, by which I mean something like the least degree of a non-trivial polynomial which vanishes on that set. One can view the polynomial method as the strategy of comparing upper and lower bounds on the Zariski complexity of sets to obtain nontrivial combinatorial consequences. I have the vague feeling that ultimately, such notions of complexity should play as prominent a role in these sorts of combinatorial problems as existing notions of "size" for such sets, such as cardinality, dimension, or Fourier uniformity.