In searching through the Riemann Nachlass in Gottingen (including those
folders not listed as connected with \zeta(s)) there is no 
evidence -- at least that has been saved -- that Riemann computed 
anything more than the first few zeros (I think up to ordinate about 80).   
The method he used was the expansion that is now called the Riemann-Siegel 
formula.  I did not see any use, e.g., of an approach based on 
Euler-Maclaurin.   The limited accuracy Riemann obtained reflects that of 
the error term in the R-S formula.