Let $\mathbb T\subset\mathbb R$ be a periodic time scale in shifts $\delta_\pm$ with period $P\in[t_0,\infty)_{\mathbb T}$. In this paper we consider the nonlinear functional dynamic equation of the form \[ x^abla(t)=a(t)x(t)-ambda b(t)f(x(h(t))),\qquad tı\mathbb T. \] By using the Krasnoselskiĭ, Avery--Henderson and Leggett--Williams fixed point theorems, we present different sufficient conditions for the nonexistence and existence of at least one, two or three positive periodic solutions in shifts $\delta_\pm$ of the above problem on time scales. We extend and unify periodic differential, difference, $h$-difference and $q$-difference equations and more by a new periodicity concept on time scales.