A novel numerical method, based on physical intuition, for particle-in-cell simulations of electromagnetic plasma microturbulence with fully kinetic ion and electron dynamics is presented. The method is based on the observation that, for low-frequency modes of interest [ω/ω_ci≪1, ω is the typical mode frequency and ω_ci is the ion cyclotron frequency] the impact of particles that have velocities larger than the resonant velocity, vr∼ω/kk (kk is the typical parallel wavenumber) is negligibly small (this is especially true for the electrons). Therefore it is natural to analytically segregate the electron response into an adiabatic response and a nonadiabatic response and to numerically resolve only the latter: this approach is termed the splitting scheme. However, the exact separation between adiabatic and nonadiabatic responses implies that a set of coupled, nonlinear elliptic equations has to be solved; in this paper an iterative technique based on the multigrid method is used to resolve the apparent numerical difficulty. It is shown that the splitting scheme allows for clean, noise-free simulations of electromagnetic drift waves and ion temperature gradient (ITG) modes. It is also shown that the advantage of noise-free kinetic simulations translates into better energy conservation properties.