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We propose a systematic way to carry out the method introduced inhep-th/0410077 for computing certain unitarity cuts of one-loop N=4 amplitudesof gluons. We observe that the class of cuts for which the method worksinvolves all next-to-MHV n-gluon one-loop amplitudes of any helicityconfigurations. As an application of our systematic procedure, we obtain thecomplete seven-gluon one-loop leading-color amplitudeA_{7;1}(1-,2-,3-,4+,5+,6+,7+).Comment: 30 pages, harvmac, 2 figures. v2: Minor corrections, typos fixed,  Appendix B added. v3: Note added at end of introduction, paragraph added in  section

Computing one-loop amplitudes from the holomorphic anomaly of unitarity cuts