I'd like to just get back to the OP's question. Valid suggestions have already been made here, but I'd like to sum it up, with a touch of my own person opinion.
If you want to abide by the laws of Japan strictly, you must abandon your Japanese citizenship after gaining US citizenship, since Japan does not allow dual citizenship. As some have mentioned, it is easy for a naturalized Japanese citizen living in Japan, who had originally been, say, American, to simply keep both Japanese and the foreign passports by just saying nothing to the Japanese authorities. This may be illegal in the strictest sense, but it is not something that is exhaustively enforced. This does not cause a problem as long as you stay a good, law-abiding citizen in Japan or wherever.
However, since you live in the US, keeping your Japanese passport poses a problem when your passport expires. When you go to the Japanese embassy to ask for extension, I suspect that they will check on your legal status in the US. If you remain a permanent resident in the US, you can keep your Japanese passport. If you are naturalized American at that point in time, you will be forced to give up Japanese citizenship by the local Japanese embassy. You will be refused issue of a Japanese passport and will be required to remove yourself from your Japanese family's registry, thus becoming a non-citizen.
Once you become a US citizen, you will be just another American in the eyes of Japanese law and will be granted only what other Americans get regarding visa, work status, etc. Working in Japan will require work visa or whatever visa you can get from the embassy. Otherwise, you can visit Japan on a "visiting relatives" visa, which I think is roughly equivalent to a tourist visa but with just a bit more leniency.
So, you will have to examine carefully what you plan to do in the future before you make your decision to gain US citizenship or remain a permanent resident. (Of course, you can simply defer becoming a US citizen until you think you want to settle in the US permanently.)
Hope it helps!!