A stochastic model for the idealized locomotion of cells is studied. The cell is assumed to cover a polygonal line in IR k, the times between turns are exponentially distributed and independent of the directions, and the density of the nth direction e does not depend on the (n - 1)th direction e'. The resulting Markov process (X(t), D(t)) for position and direction of the motion at time t is studied by using the integrodifferential equation for the transition function. For example, the joint distribution of (X(t), D(t)) is derived in closed form if n = 2 or n=3 and all chosen directions (including the initial one) are uniformly distributed. For higher dimensions the combined Fourier-Laplace transform of X(t) is given. The case of a fixed initial direction is also considered.