The CFD modeling of two-dimensional multiphase flows is a useful tool in industry, although accurate modeling itself remains a difficult task. One of the difficulties is to track the complicated topological deformations of the interfaces between different phases. This paper describes a marker-particle method designed to track fluid interfaces for fluid flows of at least three phases. The interface-tracking scheme presented in this paper is the first part of a series of papers presenting our complete model based on a one-field Godunov marker-particle projection scheme (GMPPS). In this part, we shall focus on the presentation of the interface-tracking scheme and the kinematic tests we conducted to examine the scheme’s ability to accurately track interfacial movements typified by vorticity-induced stretching and tearing of the interface. Our test results show that for a set of carefully designed and commonly used error measures, relative percentage errors never exceed 2% for all of the tests and grid sizes considered, provided a sufficient number of marker particles are used. We shall also demonstrate that the method is of second-order accuracy and the interface transition width remains constant never exceeding three cell widths.