the heir takes basis in it at its current fair market value at the time of the transfer. In the case of cash, its fair market value never changes so you'll never be taxed on it.
However, for example, if you were to inherit a house worth 100K your basis would be 100K. Then you hold it for five years and sell it for 110K. At that time you would have 10K of capital gains income. You never pay tax on the initial 100K.