The Riemann zeta function $\zeta(s)$ at complex $s$ has the statistical physics interpretation of a partition function at complex temperature. This has no direct physical meaning in general, but for certain models it does. A notable example is the Ising model, where the real and imaginary temperature axes are related by a transformation from an hexagonal to a triangular lattice.
Quite generally, the zeroes of the partition function in the complex plane fall on lines rather than in areas. For ferromagnetic models this is the content of the Yang-Lee theorem. It is therefore natural to expect the Riemann hypothesis to hold, although the Yang-Lee theorem does not cover this case.
An overview of the older literature on complex temperature partition functions is: "Location of zeros in the complex temperature plane: Absence of Lee-Yang theorem", W. van Saarloos and D. A Kurtze, J. Phys. A: Math. Gen. 17 (1984) 1301-1311. A more recent paper is "Complex-temperature partition function zeros of the Potts model on the honeycomb and kagome ́ lattices", H. Feldmann, R. Shrock, and S.-H. Tsai, Phys. Rev. E 57, 1335 (1998). There are many more papers, it is a quite active field of study.
A very recent paper is http://arxiv.org/pdf/1110.0942