This paper introduces a simple method to construct a stationary process on the real line with a Polya-type covariance function and with any infinitely divisible marginal distribution, by randomising the timescale of the increment of a second-order Levy process with an appropriate positive random variable. With the construction method extended to the multivariate case, we construct vector stochastic processes with Polya-type direct covariance functions and with any specified infinitely divisible marginal distributions. This makes available a new class of non-Gaussian vector stochastic processes with flexible correlation structure for use in modelling and simulation.