Digital chest tomosynthesis (DCT) is a technique to produce sectional 3D images of a human chest for pulmonary disease screening, with 2D X-ray projections taken within an extremely limited range of angles. However, under the limited angle scenario, DCT contains strong artifacts caused by the presence of ribs, jamming the imaging quality of the lung area. Recently, great progress has been achieved for rib suppression in a single X-ray image, to reveal a clearer lung texture. We firstly extend the rib suppression problem to the 3D case at the software level. We propose a $\textbf{T}$omosynthesis $\textbf{RI}$b Su$\textbf{P}$pression and $\textbf{L}$ung $\textbf{E}$nhancement $\textbf{Net}$work (TRIPLE-Net) to model the 3D rib component and provide a rib-free DCT. TRIPLE-Net takes the advantages from both 2D and 3D domains, which model the ribs in DCT with the exact FBP procedure and 3D depth information, respectively. The experiments on simulated datasets and clinical data have shown the effectiveness of TRIPLE-Net to preserve lung details as well as improve the imaging quality of pulmonary diseases. Finally, an expert user study confirms our findings.