This is the part that I think is interesting."ZetaGrid is a distributed computing project attempting to calculate as many zeros as possible. It had reached 1029.9 billion zeros as of Feb. 18, 2005. Gourdon (2004) used an algorithm of Odlyzko and Schönhage to calculate the first 10 x 1012 zeros (Pegg 2004, Pegg and Weisstein 2004)...All known values of t corresponding to nontrivial zeros appear to be irrational (Havil 2003, p. 195; Derbyshire 2004, p. 384)."
So for the billions of non-trivial zeroes calculated to date, the real part, t, appears to be irrational. I have to tell you that if I had the necessary knowledge, I wouldn't want to prove the hypothesis but rather I'd like to prove that the real part, t, is irrational for all non-trivial zeroes.
In the meantime, here is a rather interesting article about the Riemann Hypothesis and the number 42 (the answer to everything *grin*) - Marcus du Sautoy, "Prime Numbers Get Hitched", Seed Magazine" (03/27/2006). I found the link to this while reading about the Riemann Zeta Function on Wikipedia. It's a really cool paper talking about the link between the Riemann Hypothesis and physics. Go take a look.