One of the most promising techniques for limiting complexity for real-time 3-D ultrasound systems is to use sparse 2-D layouts. For a given number of channels, optimization of performance is desirable to ensure high quality volume images. To find optimal layouts, several approaches have been followed with varying success. The most promising designs proposed are Vernier arrays, but also these suffer from high peaks in the sidelobe region compared with a dense array. In this work, we propose new methods based on the principles of suppression of grating lobes to form symmetric and non-symmetric regular sparse periodic and radially periodic designs. The proposed methods extend the concept of sparse periodic layouts by exploiting either an increased number of symmetry axes or radial symmetry. We also introduce two new strategies to form designs with non-overlapping elements. The performance of the new layouts range from the performance of Vernier arrays to almost that of dense arrays. Our designs have simplicity in construction, flexibility in the number of active elements, and the possibility of trade off sidelobe peaks against sidelobe energy.