The nonmesonic weak decay of polarized Lambda hypernuclei is studied for the first time by taking into account, with a Monte Carlo intranuclear cascade code, the nucleon final state interactions. A one-meson-exchange model is employed to describe the Lambda-->N-->nN processes in a finite nucleus framework. The relationship between the intrinsic Lambda asymmetry parameter a(Lambda) and the asymmetry a(M)(Lambda) accessible in experiments is discussed. A strong dependence of a(M)(Lambda) on nucleon final state interactions and detection threshold is obtained. Our results for a(M)(Lambda) are consistent with (11)(Lambda)B--> and (12)(Lambda)C--> data but disagree with observations in (5)(Lambda)H-->e.