There are examples (possibly due to Hecke?) of zeta-like functions which obey almost all of the known properties of the Riemann zeta function, such as the functional equation, Euler product, and asymptotics at infinity, but which have nontrivial zeroes off the critical line. This strongly indicates that one cannot hope to prove the Riemann hypothesis purely by complex analytic methods; at some point, one needs to use something more about the integers than just the fundamental theorem of arithmetic (encoded here as the Euler product), the Poisson summation formula (encoded here as the functional equation), and asymptotic distribution in the reals (encoded here as asymptotics of zeta).
In a related spirit, there are examples (due to Diamond, Montgomery, and Vorhauer) of Beurling integers (generated by a set of Beurling primes, which have asymptotic distribution similar to that of the rational integers) whose zeta function either has non-trivial zeroes or fails to be analytically continued beyond the classical zero-free region. Admittedly this example does not have the functional equation, but it does seem to indicate that multiplicative number theory methods alone are insufficient to resolve the Riemann hypothesis.
EDIT: It turns out that my first paragraph here is based on outdated information. It is currently possible that all functions in the Selberg class (whose members obey all the axioms above, in addition to the Ramanujan conjecture) could obey the RH. I don't know though if anyone is seriously proposing that this much more general conjecture is the right way to attack RH and its relatives, though.