There is a well-known example of Davenport and Heilbronn of a Dirichlet series that in some sense is not so different from the Riemann-zeta function but that has zeros off the critical line.
The function is defined $$\sum_{n=0}^{\infty} \frac{a_n}{n^s}$$ where $a_n$ equals $1, c, -c, -1, 0$ for $n$ equal to $0,1,2,3,4$ modulo $5$, resp., with $c$ a certain algebraic number [see the reference at the end for the actual value].
This function then fulfills a functional equation similarly to the Riemann-zeta-function and (thus) can be continued to the entire plane (for details see again reference below). Yet as mentioned above it has zeros off the critical line. And, it might be worth adding that for other Dirichlet series with periodic coefficient sequences (for example, Dirichlet L-series) one expects a generalisation of RH to be true.
For some recent computational investigations on the zeros of this function see for example Zeros of the Davenport-Heilbronn Counterexample Mathematics of Computation, 2007.
For an 'axiomatic' framework where no exceptions to (the analog of) the Riemann Hypothesis are currently expected while capturing many/most Dirichlet series that appear in practise see the Selberg class.
