Significant advances have been recently made in the development of increasingly effective in-exact (or incomplete) search algorithms—particularly geared towards finding good though not provably optimal solutions fast—for the constraint optimization paradigm of maximum satisfiability (MaxSAT). One of the most successful recent approaches is a new type of stochastic local search in which a Boolean satisfiability (SAT) solver is used as a decision oracle for moving from a solution to another. In this work, we strive for extending the success of the approach to the more general realm of pseudo-Boolean optimization (PBO), where constraints are expressed as linear inequalities over binary variables. As a basis for the approach, we make use of recent advances in practical approaches to satisfiability checking pseudo-Boolean constraints. We outline various heuristics within the oracle-based approach to anytime PBO solving, and show that the approach compares in practice favorably both to a recently-proposed local search approach for PBO that is in comparison a more traditional instantiation of the stochastic local search paradigm as well as a recent exact PBO approach when used as an anytime solver.