Post-reconstruction filtering is often applied for noise suppression due to limited data counts in myocardial perfusion imaging (MPI) with single-photon emission computed tomography (SPECT). We study a deep learning (DL) approach for denoising in conventional SPECT-MPI acquisitions, and investigate whether it can be more effective for improving the detectability of perfusion defects compared to traditional postfiltering. Owing to the lack of ground truth in clinical studies, we adopt a noise-to-noise (N2N) training approach for denoising in SPECT-MPI images. We consider a coupled U-Net (CU-Net) structure which is designed to improve learning efficiency through feature map reuse. For network training we employ a bootstrap procedure to generate multiple noise realizations from list-mode clinical acquisitions. In the experiments we demonstrated the proposed approach on a set of 895 clinical studies, where the iterative OSEM algorithm with three-dimensional (3D) Gaussian postfiltering was used to reconstruct the images. We investigated the detection performance of perfusion defects in the reconstructed images using the non-prewhitening matched filter (NPWMF), evaluated the uniformity of left ventricular (LV) wall in terms of image intensity, and quantified the effect of smoothing on the spatial resolution of the reconstructed LV wall by using its full-width at half-maximum (FWHM). Compared to OSEM with Gaussian postfiltering, the DL denoised images with CU-Net significantly improved the detection performance of perfusion defects at all contrast levels (65%, 50%, 35%, and 20%). The signal-to-noise ratio (SNRD ) in the NPWMF output was increased on average by 8% over optimal Gaussian smoothing (P<10-4 , paired t-test), while the inter-subject variability was greatly reduced. The CU-Net also outperformed a 3D nonlocal means (NLM) filter and a convolutional autoencoder (CAE) denoising network in terms of SNRD . In addition, the FWHM of the LV wall in the reconstructed images was varied by less than 1%. Furthermore, CU-Net also improved the detection performance when the images were processed with less post-reconstruction smoothing (a trade-off of increased noise for better LV resolution), with SNRD improved on average by 23%. The proposed DL with N2N training approach can yield additional noise suppression in SPECT-MPI images over conventional postfiltering. For perfusion defect detection, DL with CU-Net could outperform conventional 3D Gaussian filtering with optimal setting as well as NLM and CAE.