We introduce the concept of a quasi-triply-degenerate state (QTDS) and demonstrate its relation to an effective zero refractive index (ZRI) in a two-dimensional (2D) square lattice photonic crystal (PC) of all dielectric pillars. A QTDS is characterized by a triple band structure (TBS), wherein two of the bands manifest a linear dispersion around the Γ-point, i.e. a Dirac-like cone, while the third is a flat zero refractive index (ZRI) band with a frequency that is degenerate with one of the other bands. Significantly, we find that while triple degeneracy of the bands is not observed, the three bands approach one another so close that the observable properties of PCs adapted to the QTDS frequency perform as expected of a ZRI material. We closely examine the ZRI band at the Γ-point and show that by varying the PC material and structure parameters, the ZRI band behavior extends over a wide range of dielectric refractive indices enabling materials made with polymeric constituents. Moreover, the ZRI characteristics are robust and tolerant over a range of frequencies. Furthermore, the computational screening we employ to identify QTDS parameters enables the rational design of low-loss 2D ZRI materials for a broad range of photonic applications, including distributing a common reference phase, cloaking and focusing light.