We present initial results from a Chandra survey of a complete sample of the 8 nearest (z <= 0.04) ultraluminous IR galaxies (ULIRGs), and also include the IR-luminous galaxy NGC 6240 for comparison. In this paper we use the hard X-rays (2-8 kev) to search for the possible presence of an obscured AGN. In every case, a hard X-ray source is detected in the nuclear region. If we divide the sample according to the optical/IR spectroscopic classification (starburst vs. AGN), we find that the 5 ``starburst'' ULIRGs have hard X-ray luminosities about an order-of-magnitude smaller than the 3 ``AGN'' ULIRGs. NGC 6240 has an anomalously high hard X-ray luminosity compared to the ``starburst'' ULIRGs. The Fe Kalpha line is convincingly detected in only two ULIRGs. The weakness of the Fe-K emission in these ULIRGs generally suggests that the hard X-ray spectrum is not dominated by reflection from high N_H neutral material. The hard X-ray continuum flux ranges from a few X 10^3 to a few X 10^-5 of the far-IR flux, similar to values in pure starbursts, and several orders-of-magnitude smaller than in Compton-thin AGN. The upper limits on the ratio of the Fe Kalpha to far-IR flux are below the values measured in Compton-thick type 2 Seyfert galaxies. While very large column densities of molecular gas are observed in the nuclei of these galaxies, we find no evidence that the observed X-ray sources are obscured by Compton-thick material. Thus, our new hard X-ray data do not provide direct evidence that powerful ``buried quasars'' dominate the overall energetics of most ultraluminous infrared galaxies.