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We develop methodology allowing to simulate a stationary functional time series defined by means of its spectral density operators. Our framework is general, in that it encompasses any such stationary functional time series, whether linear or not. The methodology manifests particularly significant computational gains if the spectral density operators are specified by means of their eigendecomposition or as a filtering of white noise. In the special case of linear processes, we determine the analytical expressions for the spectral density operators of functional autoregressive (fractionally integrated) moving average processes, and leverage these as part of our spectral approach, leading to substantial improvements over time-domain simulation methods in some cases. The methods are implemented as an R package (specsimfts) accompanied by several demo files that are easy to modify and can be easily used by researchers aiming to probe the finite-sample performance of their functional time series methodology by means of simulation.