I don't think there's any general algorithm that works for arbitrary semirings. The requirement to be a semiring doesn't give us a lot to work with.
However, if you have a closed semiring, then there are algorithms for solving systems of linear equations over the semiring.
A closed semiring is a semiring with a closure operator, denoted $*$, which satisfies the equation
$$a^* = 1 + a \times a^* = 1 + a^* \times a.$$
A closed semiring is also known as a star semiring.
The intuition is that $a^*$ is intended to be the sum of the infinite series
$$1 + a + a^2 + a^3 + \dots$$
For instance, the regular languages form a closed semiring under union and concatenation; the $*$ operator is the Kleene star. The real numbers form a closed semiring under addition and multiplication; the $*$ operator is $a^* = 1/(1-a)$.
Systems of linear equations over a closed semiring
Now, if you have that kind of structure, then there is an analog of Gaussian elimination. In particular, if you have a linear system of equations
$$Ax+b = x$$
where $x$ is a vector of variables over the closed semiring, $b$ is a vector of constants, and $A$ is a matrix of constants, then this has the solution
$$X = A^* B.$$
The closure operator on matrices takes a bit of work to define, but it can be computed efficiently using an analog of Gaussian elimination.
For a careful development of the theory, I recommend the following papers:
Stephen Dolan. Fun with Semirings: A functional pearl on the abuse of linear algebra. International Conference on Functional Programming, ICFP '13.
Daniel J. Lehmann. Algebraic structures for transitive closure. Theoretical Computer Science, vol 4 pp.59--76, 1977.