Ocean geochemical tracers such as radiocarbon, protactinium and thorium isotopes, and noble gases are widely used to constrain a range of physical and biogeochemical processes in both the present-day and past ocean. However their routine simulation in global ocean circulation and climate models is hindered by the computational expense of integrating them to a steady state. Here, a new approach to this long-standing ``spin-up'' problem is introduced to efficiently compute equilibrium distributions of such tracers in seasonally-forced models. Based on ``Anderson Acceleration'', a sequence acceleration technique developed in the 1960s to solve nonlinear integral equations, the new method is entirely ``black box'' and offers significant speed-up over conventional direct time integration. Moreover, it requires no preconditioning, ensures tracer conservation and is fully consistent with the numerical time-stepping scheme of the underlying model. It thus circumvents some of the drawbacks of other schemes such as matrix-free Newton Krylov that have been proposed to address this problem. An implementation specifically tailored for the batch HPC systems on which ocean and climate models are typically run is described, and the method illustrated by applying it to a variety of geochemical tracer problems. The new method, which provides speed-ups by over an order of magnitude, should make simulations of such tracers more feasible and enable their inclusion in climate change assessments such as IPCC.