Razborov-Smolensky polynomial argument on $\textrm{ACC}[q]$ where $q$ is a prime power
The result appears as Theorem 7 in the following paper by Smolensky:

Smolensky, Roman. "On representations by low-degree polynomials." Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science. IEEE, 1993. 
https://doi.org/10.1109/SFCS.1993.366874

The techniques in this paper are still algebraic but slightly different (and more general) than what people usually teach as the "Razborov-Smolensky polynomial argument".

Yuval Filmus has a nice set of notes on the above paper as well, with more exposition and details in some places:
http://www.cs.toronto.edu/~yuvalf/Smolensky.pdf 

Update: Given that Smolensky's paper is quite terse, let me say a few words about how the implied proof proceeds. Since Filmus' notes, linked above, are fairly thorough, I won't try to be completely precise, but instead give the main technical ideas.

1) The connection between $ACC[q]$ and polynomials seems to indeed be the standard one. More precisely, if there is a constant-depth circuit computing $F$, then there exists a polynomial $P$ of polylogarithmic degree that approximates $F$, where the degree depends on the depth. Unless I'm mistaken, it is actually straightforward to see that this result works even when $q$ is a prime power. See for example section 4 in these lecture notes from MIT.

2) Then, the goal is to prove that such low-degree polynomials cannot approximate the $MOD_p$ function when $(p,q) = 1$, which shows that $MOD_p \notin ACC[q]$.

3) To prove this, Smolensky establishes in Theorem 4 a relationship between 
 (i) the Hilbert function for the one-set of $F$, that is, the set $S = \{x \mid F(x) = 1\}$, and 
 (ii) the ability of any low-degree polynomial $P$ to approximate $F$, where $F = MOD_p$ for this question. 
Theorem 4 is proved in the paper (albeit in quite a terse way, hence the pointer to Filmus' notes).

4) The final step is to analyze the Hilbert function of the one-set of $F$. Smolensky refers to his proof for Parity for the "proof" of $MOD_p$ case. What actually needs to be shown is that the Hilbert function on $S$ is as large as possible, for all degrees less than $n/2$. In his words "...we proved that over $S$ there is no linear relation among monomials of degree smaller than $n/2$. Hence for $m &lt; n/2$, $l^a_m(S)$ are as large as possible." The reason he omits the formal proof is that it is indeed the same as the case for parity, once you have the equation (8) that he provides.

I hope this makes sense, with the omitted details being medium-hard exercises.