We evaluate the ability of Convolutional Neural Networks (CNNs) to predict galaxy cluster masses in the BAHAMAS hydrodynamical simulations. We train four separate single-channel networks using: stellar mass, soft X-ray flux, bolometric X-ray flux, and the Compton $y$ parameter as observational tracers, respectively... Our training set consists of $\sim$6400 synthetic cluster images generated from the simulation, while an additional $\sim$1600 images form a test set. We also train a `multi-channel' CNN by combining the four observational tracers. We utilize $\texttt{Keras}$ with a $\texttt{Tensorflow}$ backend to train the network, and all four converge within 2000 epochs. The cluster masses predicted from these networks are evaluated using the average fractional difference between predicted cluster mass and true cluster mass. The resulting predictions are especially precise for halo masses in the range $10^{13.25}M_{\odot}<M<10^{14.5}M_{\odot}$, where all five networks produce mean mass biases of order $\approx 1\%$ with a scatter on the mean bias of $\approx 0.5\%$. The network trained with Compton $y$ parameter maps yields the most precise predictions. We interpret the network's behaviour using two diagnostic tests to determine which features are used to predict cluster mass. The CNN trained with stellar mass images detect galaxies (not surprisingly), while CNNs trained with gas-based tracers utilise the shape of the signal to estimate cluster mass. (read more)