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The accretion onto the black hole in the system HDE 226868/Cygnus X-1 is powered by the strong line driven stellar wind of the O-type donor star. We study the X-ray properties of the stellar wind in the hard state of Cyg X-1 as determined with data from the Chandra High Energy Transmission Gratings. Large density and temperature inhomogeneities are present in the wind, with a fraction of the wind consisting of clumps of matter with higher density and lower temperature embedded in a photoionized gas. Absorption dips observed in the light curve are believed to be caused by these clumps. This work concentrates on the non-dip spectra as a function of orbital phase. The spectra show lines of H-like and He-like ions of S, Si, Na, Mg, Al and highly ionized Fe (Fe xvii-Fe xxiv). We measure velocity shifts, column densities, and thermal broadening of the line series. The excellent quality of these five observations allows us to investigate the orbital phase dependence of these parameters. We show that the absorber is located close to the black hole. Doppler shifted lines point at a complex wind structure in this region, while emission lines seen in some observations are from a denser medium than the absorber. The observed line profiles are phase dependent. Their shapes vary from pure, symmetric absorption at the superior conjunction to P Cygni profiles at the inferior conjunction of the black hole., Comment: A&A, accepted (26 pages, 18 figures)