We study the optimal recovery of multivariate periodic functions of the Besov class of common smoothness $SB^{\omega}_{p,\theta}$ from their values at $n$ points in terms of the quantity $R_n(SB^{\omega}_{p,\theta},L_q)$, which is a characterization of optimality of methods of recovery. The smoothness of $SB^{\omega}_{p,\theta}$ is defined via modulus of smoothness dominated by a function $\omega$ of modulus of smoothness type. With some restrictions on $\omega$ and $p, q$, we give the asymptotic order of this quantity when $n\to\infty$. An asymptotically optimal method of recovery is constructed by using the wavelet family formed from the integer translates of the dyadic scales of multivariate de la Vallée Poussin kernels.