Conventional relaxed memory ordering techniques follow a proactive model: at a synchronization point, a processor makes its own updates to memory available to other processors by executing a memory barrier instruction, ensuring that recent writes have been ordered with respect to other processors in the system. We show that this model leads to superfluous memory barriers in programs with acquire-release style synchronization, and present a combined hardware/software synchronization mechanism called conditional memory ordering (CMO) that reduces memory ordering overhead. CMO is demonstrated on a lock algorithm that identifies those dynamic lock/unlock operations for which memory ordering is unnecessary, and speculatively omits the associated memory ordering instructions. When ordering is required, this algorithm relies on a hardware mechanism for initiating a memory ordering operation on another processor. Based on evaluation using a software-only CMO prototype, we show that CMO avoids memory ordering operations for the vast majority of dynamic acquire and release operations across a set of multithreaded Java workloads, leading to significant speedups for many. However, performance improvements in the software prototype are hindered by the high cost of remote memory ordering. Using empirical data, we construct an analytical model demonstrating the benefits of a combined hardware-software implementation.