We examine the band spectrum, and associated Floquet-Bloch eigensolutions, arising in a class of three-phase periodic checkerboards. On a periodic cell [-1, 1[2, the refractive index, n, is defined by n(2) = 1 + g(1)(x(1)) + g(2)(x(2)) with g(i)(x(i)) = r(2) for 0 <= x(i) < 1, and g(i)(x(i)) = 0 for -1 <= x(i) < 0 where r(2) is constant. We find that for r(2) > -1 the lowest frequency branch goes through origin with linear behaviour, which leads to effective properties encountered in most periodic structures. However, the case whereby r(2) = -1 is very unusual, as the frequency A behaves like near the origin, where k is the wavenumber. Finally, when r(2) < -1, the lowest branch does not pass through the origin and a zero-frequency band gap opens up. In the last two cases, effective medium theory breaks down even in the quasi-static limit, while the high-frequency homogenization [R.V. Craster, J. Kaplunov, A.V. Pichugin, High-frequency homogenization for periodic media, Proc. R. Soc. Lond. Ser. A 466 (2010) 2341-2362] neatly captures the detailed features of band diagrams.