A large class of multivariate quasi-projection operators is studied. These operators are sampling-type expansions ∑_k∈Z^d c_k(f)ψ(A^j·−k), where A is a matrix and the coefficients c_k(f) are associated with a tempered distribution ψ˜. Error estimates in L_p-norm, 2 ≤ p < ∞, are obtained under the so-called weak compatibility conditions for ψ˜ and ψ, and the Strang-Fix conditions for ψ. Approximation order depends on the smoothness of f and on the order of the compatibility and the Strang-Fix conditions. A number of examples aimed at engineering applications are provided. A special attention is payed to the quasi-projection operators associated with differential-difference expansions. Applications to solving some differential-difference equations are presented.