Electrical resonators are widely used in quantum information processing, by engineering an electromagnetic interaction with qubits based on real or virtual exchange of microwave photons. This interaction relies on strong coupling between the qubits' transition dipole moments and the vacuum fluctuations of the resonator in the same manner as cavity quantum electrodynamics (QED), and has consequently come to be called 'circuit QED' (cQED). Great strides in the control of quantum information have already been made experimentally using this idea. However, the central role played by photon exchange induced by quantum fluctuations in cQED does result in some characteristic limitations. In this paper, we discuss an alternative method for coupling qubits electromagnetically via a resonator, in which no photons are exchanged, and where the resonator need not have strong quantum fluctuations. Instead, the interaction can be viewed in terms of classical, effective 'forces' exerted by the qubits on the resonator, and the resulting resonator dynamics used to produce qubit entanglement are purely classical nature. We show how this type of interaction is similar to that encountered in the manipulation of atomic ion qubits, and we exploit this analogy to construct two-qubit entangling operations that are largely insensitive to thermal or other noise in the resonator, and to its quality factor. These operations are also extensisble to larger numbers of qubits, allowing interactions to be selectively generated among any desired subset of those coupled to a single resonator. Our proposal is potentially applicable to a variety of physical qubit modalities, including superconducting and semiconducting solid-state qubits, trapped molecular ions, and possibly even electron spins in solids.