In this article, we consider the periodic event-triggered consensus problem for single-integrator multiagent systems. In existing approaches, consensus is typically achieved by trigger schemes enforcing a monotone decrease of a Lyapunov function. Such trigger schemes may, however, result in more transmissions than are actually required to ensure consensus or meet certain performance specifications. This is because a monotone decrease may be a rather conservative condition for a given Lyapunov function in an event-triggered setting. To overcome the conservativity, we propose a novel window-based trigger scheme, which allows us to leverage existing trigger schemes from the literature to derive less conservative ones. This is achieved by taking the past system behavior into account and allowing a temporary increase of the Lyapunov function as long as a decreasing tendency is still guaranteed. For that, information from previous time steps within certain time windows is considered. We provide an explicit bound on the evolution of the Lyapunov function that is the same for the window-based scheme and the (monotone) original trigger schemes. To illustrate the benefits of the window-based scheme, we validate its efficacy through a comprehensive simulation study and demonstrate that the scheme typically reduces the average update rate of the underlying communication structure in comparison to the original trigger schemes.