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A class of nonlinear simple form differential delay equations with a τ-periodic coefficient and a constant delay τ > 0 is considered. It is shown that for an arbitrary value of the period T > 4τ -- d0, for some d0 > 0, there is an equation in the class such that it possesses an asymptotically stable T-period solution. The periodic solutions are constructed explicitly for the piecewise constant nonlinearities and the periodic coefficients involved, by reduction of the problem to one-dimensional maps. The periodic solutions and their stability properties are shown to persist when the nonlinearities are "smoothed" at the discontinuity points. [ABSTRACT FROM AUTHOR]