We present the results of the analysis of neutrino observations by the Antarctic Muon and Neutrino Detector Array (AMANDA) correlated with photon observations of more than 400 gamma-ray bursts (GRBs) in the Northern Hemisphere from 1997 to 2003. During this time period, AMANDA's effective collection area for muon neutrinos was larger than that of any other existing detector. Based on our observations of zero neutrinos during and immediately prior to the GRBs in the dataset, we set the most stringent upper limit on muon neutrino emission correlated with gamma-ray bursts. Assuming a Waxman-Bahcall spectrum and incorporating all systematic uncertainties, our flux upper limit has a normalization at 1 PeV of E^2{\Phi}_{\nu} {\leq} 6.0 \times 10^{-9} GeV cm^{-2} s^{-1} sr^{-1}, with 90% of the events expected within the energy range of ~10 TeV to ~3 PeV. The impact of this limit on several theoretical models of GRBs is discussed, as well as the future potential for detection of GRBs by next generation neutrino telescopes. Finally, we briefly describe several modifications to this analysis in order to apply it to other types of transient point sources.