The Rydberg dipole blockade has emerged as the standard mechanism to induce entanglement between neutral-atom qubits. In these protocols, laser fields that couple qubit states to Rydberg states are modulated to implement entangling gates. Here we present an alternative protocol to implement entangling gates via Rydberg dressing and a microwave-field-driven spin-flip blockade [Y.-Y. Jau et al., Nat. Phys. 12, 71 (2016)]. We consider the specific example of qubits encoded in the clock states of cesium.