I've worked as contractor for companies in the U.S and Germany, normally I would just send them a receipt and received my money without problems in my bank (I did have to take care of the taxes in my country).
We have paid using SWIFT transfers, which sometimes require a bit of work to set up, but nothing too onerous - it's like doing any complicated transaction at a bank. Once set up, the recurring payments are generally trivial - as you said, we get an invoice, send payment, and things are merry.
Tax-wise this is an expense like any other to our US-based company. Note that in this case the developers actually LIVE overseas. If somebody is in the US but is not a resident / authorized to work, then it's a whole different story.