Standard Ewald sums, which calculate, e.g., the electrostatic energy or the force in periodically closed systems of charged particles, can be efficiently speeded up by the use of the fast Fourier transformation (FFT). In this article we investigate three algorithms for the FFT-accelerated Ewald sum, which have attracted widespread attention, namely, the so-called particle-particle -particle mesh ((PM)-M-3), particle mesh Ewald (PME), and smooth PME method. We present a unified view of the underlying techniques and the various ingredients which comprise those routines. Additionally, we offer detailed accuracy measurements, which shed some light on the influence of several tuning parameters and also show that the existing methods - although similar in spirit - exhibit remarkable differences in accuracy. We propose a set of combinations of the individual components, mostly relying on the (PM)-M-3 approach, that we regard to be the most flexible. The issue of estimating the errors connected with particle mesh routines is reserved to paper II.