Automatic heart sound abnormality detection can play a vital role in the early diagnosis of heart diseases, particularly in low-resource settings. The state-of-the-art algorithms for this task utilize a set of Finite Impulse Response (FIR) band-pass filters as a front-end followed by a Convolutional Neural Network (CNN) model. In this work, we propound a novel CNN architecture that integrates the front-end band-pass filters within the network using time-convolution (tConv) layers, which enables the FIR filter-bank parameters to become learnable. Different initialization strategies for the learnable filters, including random parameters and a set of predefined FIR filter-bank coefficients, are examined. Using the proposed tConv layers, we add constraints to the learnable FIR filters to ensure linear and zero phase responses. Experimental evaluations are performed on a balanced 4-fold cross-validation task prepared using the PhysioNet/CinC 2016 dataset. Results demonstrate that the proposed models yield superior performance compared to the state-of-the-art system, while the linear phase FIR filter-bank method provides an absolute improvement of 9.54% over the baseline in terms of an overall accuracy metric.