This paper proposes a periodic-modulation-based blind channel identification scheme for single-carrier (SC) block transmission with frequency-domain equalization (FDE). The proposed approach relies on the block system model and exploits the circulant channel matrix structure after the cyclic prefix is removed. It is shown that the set of linear equations relating the autocorrelation matrix of the block received signal and the product channel coefficients can be rearranged into one with a distinctive block circulant structure. The identification equations thus obtained lead to a very simple identifiability condition, as well as a natural formulation of the optimal modulating sequence design problem which, based on the block circulant signal structure, can be cast as a constrained quadratic problem that allows for a simple closed-form solution. The impact of the optimal modulating sequence on the peak-to-average power ratio (PAPR) is investigated. Also, it is shown that the optimal sequence results in a consistent channel estimate irrespective of white noise perturbation. Pairwise error probability analysis is used to examine the equalization performance and based on which several design tradeoffs are discussed. Simulation results are used for illustrating the performance of the proposed method.