[I'm posting this as an answer despite being in $\mathsf{PromiseMA}$ because a) it exhibits a different kind of $\mathsf{MA}$-style algorithm, and b) @PeterShor asked and it was too long for a comment.]
Over any finite field $\mathbb{F}$, the following problem is in $\mathsf{PromiseMA}$:
Input: A set of $\mathbb{F}$-polynomials $F_1(\vec{x}), \dotsc, F_m(\vec{x})$
Decide: Is there no solution to $F_1(\vec{x}) = \dotsb = F_m(\vec{x}) = 0$ over the algebraic closure $\overline{\mathbb{F}}$
Promise: Either there is a solution over $\overline{\mathbb{F}}$, or there is a poly-size $\mathbb{F}$-algebraic circuit $C(\vec{x}, y_1, \dotsc, y_m)$ such that $C(\vec{x}, \vec{0}) = 0$ and $C(\vec{x}, \vec{F}(\vec{x})) = 1$ (identically as polynomials)
The $\mathsf{MA}$-style algorithm guesses the circuit $C$, and then verifies the two conditions using polynomial identity testing, which is in $\mathsf{coRP}$ by Schwarz-Zippell-DeMillo-Lipton.
Note that, without the restriction of poly-size, Hilbert's Nullstellensatz guarantees that there is no solution if and only if there is some circuit $C$ satisfying the above two conditions. (On the other hand, assuming $\mathsf{NP} \not\subseteq \mathsf{coMA}$, there are systems of equations coming from algebrizations of 3SAT for which the above poly-size promise is violated.)
(This is the basis of an algebraic proof system from recent joint work with Toniann Pitassi, but for the purposes of this answer similar ideas go back to an earlier paper of Pitassi's as well as her 1998 ICM talk, and to the so-called Nullstellensatz and Polynomial Calculus proof systems.)
