Multivariate quasi-projection operators Q _j(f,φ,φ˜), associated with a function φ and a distribution/function φ˜, are considered. The function φ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with φ˜. Using technique based on the Fourier multipliers, we study approximation properties of such operators for functions f from anisotropic Besov spaces and L _p spaces with 1≤p≤∞. In particular, upper and lower estimates of the L _p-error of approximation in terms of anisotropic moduli of smoothness and anisotropic best approximations are obtained.