You simply travel on your green card and the passport of your country. So long as you are not gone for longer than 6 months you will have no trouble returning to the United States.
Charles H. Kuck
Kuck Immigration Partners LLC--The Immigration Law Firm
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In addition to the answer above, you have to check Japan's visa and entry requirements for your country of citizenship. Because you are not a citizen of the United Stated, the regulations governing entry regimen between Japan and you native country will apply even though you are traveling from the US.