Using means per months to control for monthly time effects

I would like to have some advice on a simple topic. Controlling for time effects and removing seasonality can be done with many ways, like for example using a 12th lag in yearly data to control for seasonality in arima framework, or using time dummies in panel data for the same reason. I wonder if it is correct to use averages across months and include them in a regression equation to control for potential seasonality in the same way we use panel dummies to control for panel specific effects. My thinking is that, as we use panel dummies to an OLS regression and have same results with fixed effects (fixed effects come from subtraction of panel average across time), so we can use monthly averages across panels as a regressor (or subtract them from each observation) to emulate the effect of using time dummies. My problem is that my sample has too many periods to include time dummies in the model, while at the same time, de-seasonalizing with excel seems too naive method. Thank you in advance.