By leveraging the waveform superposition property of the multiple access channel, over-the-air computation (AirComp) enables the execution of digital computations through analog means in the wireless domain, leading to faster processing and reduced latency. In this paper, we propose a novel approach to implement a neural network (NN) consisting of digital fully connected (FC) layers using physically reconfigurable hardware. Specifically, we investigate reconfigurable intelligent surfaces (RISs)-assisted multiple-input multiple-output (MIMO) systems to emulate the functionality of a NN for over-the-air inference. In this setup, both the RIS and the transceiver are jointly configured to manipulate the ambient wireless propagation environment, effectively reproducing the adjustable weights of a digital FC layer. We refer to this new computational paradigm as \textit{AirFC}. We formulate an imitation error minimization problem between the effective channel created by RIS and a target FC layer by jointly optimizing over-the-air parameters. To solve this non-convex optimization problem, an extremely low-complexity alternating optimization algorithm is proposed, where semi-closed-form/closed-form solutions for all optimization variables are derived. Simulation results show that the RIS-assisted MIMO-based AirFC can achieve competitive classification accuracy. Furthermore, it is also shown that a multi-RIS configuration significantly outperforms a single-RIS setup, particularly in line-of-sight (LoS)-dominated channels.