Particle filters are key algorithms for object tracking under non-linear, non-Gaussian dynamics. The high computational cost of particle filters, however, hampers their applicability in cases where the likelihood model is costly to evaluate, or where large numbers of particles are required to represent the posterior. We introduce the piecewise constant sequential importance sampling/resampling (pcSIR) algorithm, which aims at reducing the cost of traditional particle filters by approximating the likelihood with a mixture of uniform distributions over pre-defined cells or bins. The particles in each bin are represented by a dummy particle at the center of mass of the original particle distribution and with a state vector that is the average of the states of all particles in the same bin. The likelihood is only evaluated for the dummy particles, and the resulting weight is identically assigned to all particles in the bin. We derive upper bounds on the approximation error of the so-obtained piecewise constant function representation, and analyze how bin size affects tracking accuracy and runtime. Further, we show numerically that the pcSIR approximation error converges to that of sequential importance sampling/resampling (SIR) as the bin size is decreased. We present a set of numerical experiments from the field of biological image processing and tracking that demonstrate pcSIR's capabilities. Overall, we consider pcSIR a promising candidate for simple, fast particle filtering in generic applications, especially in those with a costly likelihood update step.