In this paper, the average consensus problem for multi-agent systems is addressed. A novel event-based control strategy is proposed which renders both control signals and state measurements, which are broadcast over the network, piecewise constant. This enables implementation on digital platforms such as microprocessors. Different triggering conditions guaranteeing convergence to an adjustable region around the average consensus point or asymptotic convergence to this point, respectively, are discussed. Numerical simulations show the effectiveness of this approach, outperforming traditional time-scheduled control in terms of load on the communication medium. Both single- and double-integrator agents are covered.