Data can have trends, cycles, seasonal pattern repeats and irregular components. How do these components impact on the ordinary multiple regression with dummy variables?
The short answer is "enormously". If you have time series data that you want to fit a regression to, you will need at a minimum to deal with the seasonality and with autocorrelation between residuals (because the data are related over time, each datapoint cannot be treated as an independent observation - the closer they are related over time, the less new information there is in each observation).
Failing to do either of these will certainly mean your regression returns misleading results. Normally you will get false positives; but you could easily also miss genuine structure that is masked (for example) by the seasonality.
There are packages that allow you to deal with autocorrelated errors in the multiple regression context eg the nlme library in R. There is a big range of methods of dealing with seasonality.
Browsing through the various questions on this site with seasonality or time-series tags gives a sense of some (far from all) of the issues. To really address this you need to look into time series analysis in some detail. Just dealing with seasonality is a vast topic...