We observed the bright phase of the 2003 outburst of the Galactic black hole candidate H 1743-322 in X-rays simultaneously with Chandra and RXTE on four occasions. The Chandra/HETGS spectra reveal narrow, variable (He-like) Fe XXV and (H-like) Fe XXVI resonance absorption lines. In the first observation, the Fe XXVI line has a FWHM of 1800 +/- 400 km/s and a blue-shift of 700 +/- 200 km/s, suggesting that the highly ionized medium is an outflow. Moreover, the Fe XXV line is observed to vary significantly on a timescale of a few hundred seconds in the first observation, which corresponds to the Keplerian orbital period at approximately 1 E+4 gravitational radii. Our models for the absorption geometry suggest that a combination of geometric effects and changing ionizing flux are required to account for the large changes in line flux observed between observations, and that the absorption likely occurs at a radius less than 1 E+4 radii for a 10 Msun black hole. Viable models for the absorption geometry include cyclic absorption due to an accretion disk structure, absorption in a clumpy outflowing disk wind, or possibly a combination of these two. If the wind in H 1743-322 has unity filling factor, the highest implied mass outflow rate is 20 percent of the Eddington mass accretion rate. This wind may be a hot precursor to the Seyfert-like, outflowing warm absorber geometries recently found in the Galactic black holes GX 339-4 and XTE J1650-500. We discuss these findings in the context of ionized Fe absorption lines found in the spectra of other Galactic sources, and connections to warm absorbers, winds, and jets in other accreting systems.