Multivariate factor stochastic volatility (SV) models are increasingly used for the analysis of multivariate financial and economic time series because they can capture the volatility dynamics by a small number of latent factors. The main advantage of such a model is its parsimony, as the variances and covariances of a time series vector are governed by a low-dimensional common factor with the components following independent SV models. For high-dimensional problems of this kind, Bayesian MCMC estimation is a very efficient estimation method; however, it is associated with a considerable computational burden when the dimensionality of the data is moderate to large. To overcome this, we avoid the usual forward-filtering backward-sampling (FFBS) algorithm by sampling "all without a loop" (AWOL), consider various reparameterizations such as (partial) noncentering, and apply an ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation at a univariate level, which can be applied directly to heteroskedasticity estimation for latent variables such as factors. To show the effectiveness of our approach, we apply the model to a vector of daily exchange rate data.