By the first-principles electronic structure calculations, we find that the ground state of PbO-type tetragonal alpha-FeTe is in a bicollinear antiferromagnetic order, in which the Fe local moments (approximately 2.5 microB) align ferromagnetically along a diagonal direction and antiferromagnetically along the other diagonal direction on the Fe square lattice. This novel bicollinear order results from the interplay among the nearest, the next-nearest, and the next-next-nearest neighbor superexchange interactions, mediated by Te 5p band. In contrast, the ground state of alpha-FeSe is in a collinear antiferromagnetic order, similar to those in LaFeAsO and BaFe2As2. This finding sheds new light on the origin of magnetic ordering in Fe-based superconductors.