The C-H stretching region looks a lot like polypropylene to my eye- certainly it is representative of a hydrocarbon that has a good mix of both CH2 and CH3 groups (i.e., something highly branched or relatively short chain). You can compare it to the spectrum of polyethylene which mostly just has two sharp peaks owing to almost all C's being CH2. The sharp peaks all occur below 1600 wavenumbers, which except in some unusual circumstances rules out most carbonyls. Amides are pretty low as far as carbonyls go but not below 1600 cm-1 typically. Plus I'm not sure what grease would have an amide group. The sharp peaks below 1600 cm-1 are consistent with scissoring and other bending modes of CH2 and CH3 groups, which would be plentiful in a grease, although they're not often so close to 1600 cm-1 (usually more in the 1400-1500 cm region). 1500-1600 is the right region of aromatic ring stretches.
I'm guessing the noisy region below 1000 cm-1 is just poor absorptivity of your instrument? This region would have most of your low frequency aliphatic rocking motions.
What kind of grease is it? Definitely not a silicon-based grease, due to total absence of Si-H stretches.