MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer in this paper an approach to reduce the computational costs of such algorithms by a simple and universal divide-and-conquer strategy. The idea behind the generic acceleration is to divide the acceptance step into several parts, aiming at a major reduction in computing time that outranks the corresponding reduction in acceptance probability. The division decomposes the “prior x likelihood” term into a product such that some of its components are much cheaper to compute than others. Each of the components can be sequentially compared with a uniform variate, the first rejection signalling that the proposed value is considered no further, This approach can in turn be accelerated as part of a prefetching algorithm taking advantage of the parallel abilities of the computer at hand. We illustrate those accelerating features on a series of toy and realistic examples.