How to make the Black-Sholes model work with a company that pays dividends?

The no dividends assumption is actually not one of the fundamental assumptions of the BS model.

You have to decide whether or not you want to model the dividends continuously or discretely. This is discussed in the Wikipedia entry for Black Scholes under "Extensions of the model." Like it says, using a continuously compounded dividend rate is typically better suited for modeling an investment basket having a large number of stocks, whereas if you have a single stock with known payout dates, then you'll probably want to factor in each dividend payment discretely.