Persian,
While fish oil has recognized heart related benefits and does potentially increase the risk of spontaneous bleeding (in doses over 3 grams per day), it should not be substituted for regulated coumadin use.
Coumadin is standardized for dosage, potency, and quality. There are mountains of information regarding coumadin use for the purpose of anti-coagulation in the presence of a mechanical heart valve.
There are no such studies and certainly no long term evidence that fish oil is in any way an effective substitute for prescribed coumadin as an anti-coagulant for a person with a mechanical heart valve.
Please DO NOT attempt to take fish oil as a substitute for coumadin. You would be placing yourself at increased risk of blood clots which could lead to stroke.
Please also consult your physician before adding fish oil supplements to your diet since it can have an additive affect with coumadin (and other drugs such as statins or insulin).