A new approximate expression for gamma-ray buildup factors of multilayered shields is proposed. The expression is formulated based on the vector form and considers the gamma-ray energy spectrum directly. It treats the gamma-ray transmission by a transmission matrix and the backscattering by an albedo matrix. Its capability of reproducing the buildup factors for multilayered shields is demonstrated by using double-layered shields composed of two materials of water, iron, and lead at 1 and 10 MeV. The data of three-layered shields of these materials are also very well reproduced. The mechanism of the density effect arising, which appears in the buildup factor for a point isotropic source, is clearly interpreted by the current method to be a geometrical effect. A correction factor for incorporating the density effect into the current expression is derived. The modified expression is successfully applied to buildup factors for a 0.5-MeV point isotropic source for two-layered shields of water and iron.