We present the results of searches for high-energy muon neutrinos from 41 gamma-ray bursts (GRBs) in the northern sky with the IceCube detector in its 22 string configuration active in 2007/2008. The searches cover both the prompt and a possible precursor emission as well as a model-independent, wide time window of -1 hr to + 3 hr around each GRB. In contrast to previous searches with a large GRB population, we do not utilize a standard Waxman-Bahcall GRB flux for the prompt emission but calculate individual neutrino spectra for all 41 GRBs from the burst parameters measured by satellites. For all of the three time windows, the best estimate for the number of signal events is zero. Therefore, we place 90% CL upper limits on the fluence from the prompt phase of 3.7 x 10(-3) erg cm(-2) (72 TeV-6.5 PeV) and on the fluence from the precursor phase of 2.3 x 10(-3) erg cm(-2) (2.2-55 TeV), where the quoted energy ranges contain 90% of the expected signal events in the detector. The 90% CL upper limit for the wide time window is 2.7 x 10(-3) erg cm(-2) (3 TeV-2.8 PeV) assuming an E-2 flux.