The basic idea is that wavelet coefficients of real signals typically exhibit clustering patterns, in that they contain connected regions of coefficients of similar magnitude (large or small). We present a Bayesian approach to wavelet de-noising, that exploits this dependence between neighbouring wavelet coefficients by a priori modelling them via a Markov chain-based prior. We call this prior the caravan prior. Posterior computations in our method are performed via the Gibbs sampler. Using representative synthetic and real data examples, we conduct a detailed comparison of our approach with a benchmark empirical Bayes de-noising method due to Johnstone and Silverman, see here and here. We show that the caravan prior fares well and is therefore a useful addition to the wavelet de-noising toolbox.