Lightweight deep learning models are increasingly required in resource-constrained environments such as mobile devices and the Internet of Medical Things (IoMT). Multi-head convolution with channel attention can facilitate learning activations relevant to different kernel sizes within a multi-head convolutional layer. Therefore, this study investigates the capability of novel lightweight models incorporating residual multi-head convolution with channel attention (ResMHCNN) blocks to classify medical images. We introduced three novel lightweight deep learning models (BT-Net, LCC-Net, and BC-Net) utilizing the ResMHCNN block as their backbone. These models were cross-validated and tested on three publicly available medical image datasets: a brain tumor dataset from Figshare consisting of T1-weighted magnetic resonance imaging slices of meningioma, glioma, and pituitary tumors; the LC25000 dataset, which includes microscopic images of lung and colon cancers; and the BreaKHis dataset, containing benign and malignant breast microscopic images. The lightweight models achieved accuracies of 96.9% for 3-class brain tumor classification using BT-Net, and 99.7% for 5-class lung and colon cancer classification using LCC-Net. For 2-class breast cancer classification, BC-Net achieved an accuracy of 96.7%. The parameter counts for the proposed lightweight models—LCC-Net, BC-Net, and BT-Net—are 0.528, 0.226, and 1.154 million, respectively. The presented lightweight models, featuring ResMHCNN blocks, may be effectively employed for accurate medical image classification. In the future, these models might be tested for viability in resource-constrained systems such as mobile devices and IoMT platforms.