Although anticoagulants like heparin and warfarin do not remove the existing clots, they may keep them from growing, while your body fights the clot. This will subsequently control the acute symptoms, prevent a major pulmonary embolism and reduce the severity of post-thrombotic syndrome (leg swelling, heaviness, leg ulcers).
In the acute period patients will receive heparin (intravenous or under the skin (subcutaneous) and they will then need transition to warfarin (Coumadin) that is taken orally daily.
Treatment with blood thinners may last from three to six months, but depending on your medical history it may need to be lifelong.
A major disadvantage of Coumadin is its interaction with certain foods (mainly green vegetables) and the need for frequent blood tests to check the appropriateness of the dosage (too little increases your clot risk, too much increases your risk for bleeding).
New medications are currently available (e.g. dabigatran (Pradaxa), apixaban (Eliquis), rivaroxaban (Xarelto)) that may not have these disadvantages, they may however carry other risks (e.g. no antidote available to cease bleeding in case of injury). The most common side effect of all blood-thinners is bleeding.
You should discuss these options with your doctor.