Most of the existing single-stage and two-stage 3D object detectors are anchor-based methods, while the efficient but challenging anchor-free single-stage 3D object detection is not well investigated. Recent studies on 2D object detection show that the anchor-free methods also are of great potential. However, the unordered and sparse properties of point clouds prevent us from directly leveraging the advanced 2D methods on 3D point clouds. We overcome this by converting the voxel-based sparse 3D feature volumes into the sparse 2D feature maps. We propose an attentive module to fit the sparse feature maps to dense mostly on the object regions through the deformable convolution tower and the supervised mask-guided attention. By directly regressing the 3D bounding box from the enhanced and dense feature maps, we construct a novel single-stage 3D detector for point clouds in an anchor-free manner. We propose an IoU-based detection confidence re-calibration scheme to improve the correlation between the detection confidence score and the accuracy of the bounding box regression. Our code is publicly available at https://github.com/jialeli1/MGAF-3DSSD.