The methodology implemented in the software X-12, X-13 ARIMA and TRAMO-SEATS can be considered the standard practice in Statistical Offices such as as Eurostat, the US Census Bureau and others. These programs will give you much more information and, in general, a better result than the use of seasonal dummies that you mention.
Both programs can be called from the command-line. It's possible to create a script that runs the program for a large number of time series and stores the results for each series.
Default options are available and can be suitable when working with a large number of series. For example, if the parameter
RSA=3 is used in TRAMO-SEATS (excerpt from the documentation): The program tests for the log/level specification, interpolates missing observations (if any), and performs automatic model identification and outlier detection. [...] The full output file is produced. X-12-ARIMA can be run with default options as well. In both cases, a text file can be created in order to choose the default procedure or other parameters of the procedure.
Unobserved components (UC) models mentioned by @forecaster are an interesting alternative. The R packages stsm and tsoutliers can be used to perform an analysis similar to TRAMO-SEATS in the context of the basic structural time series model (trend plus seasonal UC model). These packages are relatively new and have not been tested and polished as further as the other tools mentioned before.
As mentioned by @RichardHardy, loess smoothing is another alternative. This is a non-parametric approach, so you won't get in return a fitted time series model, which may be useful as a neat representation of the overall dynamics of the data, to describe or test some of the features of the data or to get standard errors for the trend or seasonal components. But if your goal is to get a smoothed series free of seasonal fluctuations you could try this approach as well.