The authors study semicirculant preconditioners \(M\) used in CG-like iterative methods for solving linear systems \(Bu=b\) of \(n\) algebraic equations arising typically from the implicit time and finite difference spatial discretizations of initial-boundary value problems for linear systems of partial differential equations of the form \(\partial u/\partial t={\mathfrak R}u\), where \(\mathfrak R\) is a spatial, 1st-order differential operator. In such cases, the matrix \(B\) is typically non-symmetric and block- tridiagonal periodic. The blocks on the main diagonal are tridiagonal periodic, and each entry in the blocks is a small matrix with arbitrary structure. The dimension of these small matrices is equal to the number of unknown functions in the system of partial differential equations. Similarly, the lower and upper diagonal blocks are block diagonal with small matrices on the diagonal. The preconditioning system \(Mw=r\) arising at each iteration step can be solved very efficiently via the fast Fourier transform with the cost of \(O(n\log(n))\) arithmetical operations. The spectrum of \(M^{-1}B\) is studied theoretically and numerically. Under certain assumptions, the spectrum is asymptotically bounded and has clustering properties. Thus, the preconditioned CG-like solver must be very efficient. This is confirmed by the presented numerical experiment.