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 using 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.