We have developed a simulation method with both electrons and ions represented as particles-in-cell, in which the electrostatic field is determined from the requirement of quasineutrality rather than from Poisson's equation. This approach permits self-consistent calculation of the potential, in quasineutral situations where statistical fluctuations in the charge density frustrate the use of Poisson's equation. Time steps may be orders of magnitude longer than the plasma period, and mesh cells orders or magnitude longer than the Debye length, since electron plasma oscillations do not appear in the model and the Debye length is essentially set to zero. The simulation technique correctly represents kinetic features such as electron and ion Landau damping. The method is demonstrated by application to several simple test problems, including free expansion of a plasma, and linear and nonlinear ion sound. In the case of a plasma with strongly magnetized electrons, we apply the technique to determine the parallel electric field and parallel transport within the plasma. Quasineutral techniques for representing cross-field transport, and edge effects in bounded plasmas, will be discussed in subsequent publications.