For even $N\ge2$ and $\delta\ge2N-3$(for $N=2,\text{or}4$ we assume that $\delta>(N-1)/2$) we find asymptotic approximations for the quantity $$ E_R^\delta(H_{\rm N}^\omega)=\mathop{sup}\limits_{f\in H_{\rm N}^\omega}\|f(x)-S_R^\omega(x,f)\|_C(R\to\infty),$$ where $S_R^\delta(x,f)$ the spherical Riesz mean of order delta of the Fourier kernel of the function $f(x)$, and $H_N^\omega$ is the class of periodic functions of $N$ variables whose moduli of continuity do not exceed a given convex modulus of continuity $\omega(\delta)$. For $N=2$ and $\delta>1/2$ the result is known.