Well, suppose pi is a cuspidal automorphic representation of GL(n)/Q. This has the structure of a tensor product, indexed by primes p, of representations pi_p of the groups GLn(Qp). The Satake isomorphism tells us that at almost all primes, each pip is determined by a conjugacy class A(p) in GLn(C). In this language, the Riemann hypothesis for the L-function associated to pi says that the partial sums of tr(A(p)) over p < X show "as much cancellation as possible," and are of size sqrt(X). But if n>1, we are dealing with very complicated objects, and the local components of these automorphic representations vary in some incomprehensible way...
You are right, there are certainly special cases. If we knew GRH for L-functions associated to Artin representations then the Cebotarev density theorem would follow with an optimal error term. Likewise, GRH for all the symmetric powers of a fixed elliptic curve E implies (and is in fact equivalent to; see Mazur's BAMS article for a reference) the Sato-Tate conjecture for E with an optimal error term. But in general, reformulations like this simply don't exist.
There are many interesting consequences of GRH for various families of automorphic L-functions. I recommend Iwaniec and Kowalski's book (Chapter 5), the paper "Low-lying zeros of families of L-functions" by Iwaniec-Luo-Sarnak, and Sarnak's article at http://www.claymath.org/millennium/Riemann_Hypothesis/Sarnak_RH.pdf
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
$$\sum_{n=1}^{\infty} \frac{a_n}{n^s}$$
where $a_n$ equals $1, c, -c, -1, 0$ for $n$ equal to $1,2,3,4,5$ 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 (nontrivial) 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.
Best Answer
The Riemann hypothesis is proved over function fields (like the fraction field of F_q[t]), not finite fields, and the "real version" is a question about the integers. Kakeya is proved over finite fields, and the "real version" is a question about, well, the reals. So the situations are quite different. I'd say that the truth of RH over function fields really does make me feel more confidence that RH is true over number fields, because the analogy between function fields and number fields is in many ways a very close one. On the other hand, the truth of Kakeya over finite fields does not tell me very much about the truth of the real Kakeya problem. For one thing, what's true over finite fields -- that a Kakeya set has measure bounded away from 0 -- is totally false over the real numbers! (See: Besicovich set.) An intermediate multiple-scale case is that of a power series ring over a finite field -- in this case, measure-0 Kakeya sets exist by a theorem of Dummit and Hablicsek, but we don't know the answer to the Kakeya problem, i.e. we don't know whether a Kakeya set over F_q[[t]] has full dimension.
One reason (but not the only reason) the Kakeya problem is easier over finite fields than it is over the real numbers is that finite fields only have one scale, while the real numbers have multiple scales.
One reason (but not the only reason) the Riemann Hypothesis is easier over function fields than it is over number fields is that the zeta function in the function field case has an interpretation in terms of the etale cohomology of a variety over a finite field. In the number field case there is no such interpretation at present, despite the best efforts of the F_un-ologists.