Using the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lamé potentials ma(a+1)sn2(x,m) are computed for integer values a=1,2,3,… . For all cases (except a=1), we show that the partner potential is distinctly different from the original Lamé potential, even though they both have the same energy band structure. We also derive and discuss the energy band edges of the associated Lamé potentials pm sn2(x,m)+qm cn2(x,m)/dn2(x,m), which constitute a much richer class of periodic problems. Computation of their supersymmetric partners yields many additional new solvable and quasiexactly solvable periodic potentials.