Deoxygenation of triglycerides (e.g., vegetable and microalgae oils) through hydro-upgrading is one of the most important routes to produce oxygen-free hydrocarbon fuels from biomass. Even though natural triglycerides are composed of fatty acids of different structures, their effects on catalytic selectivity and deactivation during deoxygenation have not been comprehensively investigated. In this study, we used three different reactant triglycerides including palm oil, soybean oil, and linseed oil for catalytic deoxygenation over Pt/gamma-Al2O3. These oils consist of fatty acids with increasing degrees of unsaturation in the order of palm oil (number of C=C per fatty acid: 0.55) < soybean oil (1.33) < linseed oil (2.18). The catalytic results showed that the increased unsaturation of the triglycerides caused enhanced formation of heavy products by oligomerization (mostly dimers) of unsaturated fatty acid derivatives through Diels Alder reaction and/or radical addition. These heavy products were readily transformed to coke species, causing catalyst deactivation. In this regard, we demonstrated that the prehydrogenation of triglycerides at a mild temperature is highly beneficial for steadily obtaining a high yield of diesel-range paraffins and retarding catalyst deactivation in deoxygenation.