Dirac-like electronic states are the main engines powering tremendous advancements in the research of graphene, topological insulators and other materials with these states. Zero effective mass, high carrier mobility and numerous applications are some consequences of linear dispersion that distinguishes Dirac states. Here we report a new class of linear electronic bands in two-dimensional materials with zero electron effective mass and sharp band edges, and predict stable materials with such electronic structures utilizing symmetry group analysis and an ab initio approach. We make a full classification of completely linear bands in two-dimensional materials and find that only two classes exist: Dirac fermions on the one hand and fortune teller-like states on the other hand. The new class supports zero effective mass similar to that of graphene and anisotropic electronic properties like that of phosphorene.