Of course, it depends on the nature and size of your company. But as a rule of thumb, you probably don't need to open a bank account in a country until you have a local office there.
As far as working with banks with branches in multiple countries go, the few reports I've heard haven't been good. If you have a US-based account at one of the big banks, and you walk into one of their branches in Tokyo, you can't count on being able to make a deposit or sort out problems with the account. Also, the US folks at those banks are often ignorant about what their foreign branches can and can't do. So if you decide to go that route, you need to ask about the services you need, and take anything you hear with a big grain of salt until you've actually had things succeed.
I've done hundreds of thousands of dollars of international business, and my parents have done millions, all through mid-sized regional US banks. You just have to find a bank that's clueful about things like sending and receiving international wire transfers, and won't nickel and dime you to death on the fees.
I would keep it simple and have the money wired to your account in your home country. That way, everything is in one place.
Having multiple accounts, all over the world, is cumbersome and not really worth it.
You can also get a local payment processor that can act as a clearing house for transactions. That might be a better way to go instead of all those different bank accounts.