Distributed multiple-input multiple-output (D-MIMO) is a promising technology for simultaneous communication and positioning. However, phase synchronization between multiple access points in D-MIMO is challenging and methods that function without the need for phase synchronization are highly desired. Therefore, we present a method for D-MIMO that performs direct positioning of a moving device based on the delay-Doppler characteristics of the channel state information (CSI). Our method relies on particle-filter-based Bayesian inference with a state-space model. We use recent measurements from a sub-6 GHz D-MIMO OFDM system in an industrial environment to demonstrate near-centimeter accuracy under partial line-of-sight (LoS) conditions and decimeter accuracy under fully obstructed LoS.