We consider reaction-diffusion equations $u_t=\Delta u+f(u)$ on the entire space $\mathbb R^N$, $N\ge 4$. Assuming that the function $f$ is sufficiently smooth ($C^2$ is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions $u(x,t)$ which are radial in $x$, and periodic and nonconstant in $t$. We also prove some weaker nonexistence results for $N=3$. In dimensions $N=1,2$, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.