This paper introduces two new procedures for the construction of non-directional two-dimensional neighbor designs which can be used in agriculture, biometrics, and other fields. The first method that we introduce generates equi-neighbor balanced designs and partially neighbor balanced designs for bordered and un-bordered plots simultaneously. The second method generates Latin squares and partially neighbor balanced designs using properties of primitive roots. The proposed designs are balanced and partially balanced with respect to nearest neighbors in any of four main directions, north, south, east and west, say. Our constructed designs are recommended for the cases where the performance of treatment is affected by its immediate neighbors.