It is well known that for each strictly monotone list of n real numbers, there are infinitely many Jacobi matrices of order n whose eigenvalues are the entries of the list, along with a unique bisymmetric Jacobi matrix that satisfies this property. Families of such of matrices, with known spectra, are often used to test algorithms for eigenvalue determination or for recovering matrices from spectral data. Unfortunately, there are very few families of Jacobi matrices, even among bisymmetric ones, whose spectrum is known. In fact, for the specific class of Jacobi matrices with periodic coefficients, to our knowledge, the bisymmetric case has not been treated except for a few very particular examples. In this work, we seek to establish a general procedure for determining the spectral characteristics, eigenvalues, and eigenvectors of Jacobi matrices with periodic coefficients. We will then specialize the obtained formulas to the bisymmetric case for lower periods, up to 2. Our technique is based on the analysis of difference equations with periodic coefficients and the associated boundary problems.