Abstract: The newly discovered intercalated ternary iron-selenide superconductors AyFexSe2 (A=K, Rb, Cs, and/or Tl) show novel phenomena. To clarify these phenomena, we have carried out first-principles electronic structure calculations on AFe2Se2 with perfect tetragonal FeSe layers, AFe1.5Se2 with one-quarter Fe vacancies ordered in the 4×2 or 2×2 superstructure, and A0.8Fe1.6Se2 with one-fifth Fe vacancies in the 5 × 5 superstructure. We find that the AFe2Se2 compounds are bi-collinear antiferromagnetic semimetals, while both AFe1.5Se2 and A0.8Fe1.6Se2 are antiferromagnetic semiconductors with band gaps of dozens and hundreds of meV, respectively; their ground states are in a collinear antiferromagnetic order and blocked checkerboard antiferromagnetic order, respectively. The underlying physics is discussed.