We present infrared L-band (3-4 micron) nuclear spectra of a large sample of nearby ultraluminous infrared galaxies (ULIRGs).ULIRGs classified optically as non-Seyferts (LINERs, HII-regions, and unclassified) are our main targets. Using the 3.3 micron polycyclic aromatic hydrocarbon (PAH) emission and absorption features at 3.1 micron due to ice-covered dust and at 3.4 micron produced by bare carbonaceous dust, we search for signatures of powerful active galactic nuclei (AGNs) deeply buried along virtually all lines-of-sight. The 3.3 micron PAH emission, the signatures of starbursts, is detected in all but two non-Seyfert ULIRGs, but the estimated starburst magnitudes can account for only a small fraction of the infrared luminosities. Three LINER ULIRGs show spectra typical of almost pure buried AGNs, namely, strong absorption features with very small equivalent-width PAH emission. Besides these three sources, 14 LINER and 3 HII ULIRGs nuclei show strong absorption features whose absolute optical depths suggest an energy source more centrally concentrated than the surrounding dust, such as a buried AGN. In total, 17 out of 27 (63%) LINER and 3 out of 13 (23%) HII ULIRGs nuclei show some degree of evidence for powerful buried AGNs, suggesting that powerful buried AGNs may be more common in LINER ULIRGs than in HII ULIRGs. The evidence of AGNs is found in non-Seyfert ULIRGs with both warm and cool far-infrared colors. These spectra are compared with those of 15 ULIRGs nuclei with optical Seyfert signatures taken for comparison.The overall spectral properties suggest that the total amount of dust around buried AGNs in non-Seyfert ULIRGs is systematically larger than that around AGNs in Seyfert 2 ULIRGs.