We present initial results from a Chandra survey of a complete sample of the eight 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 active galactic nucleus (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 five "starburst" ULIRGs have hard X-ray luminosities about an order of magnitude smaller than the three "AGN" ULIRGs. NGC 6240 has an anomalously high hard X-ray luminosity compared to the starburst ULIRGs. The Fe-Kα 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 × 10^-3 to a few × 10^-5 of the far-IR flux, similar to values in pure starbursts and several orders of magnitude smaller than in Compton-thin AGNs. The upper limits on the ratio of the Fe-Kα-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 ULIRGs.