I know that there are some centres that advertise specific affiliations, but I have no experience of those. Perhaps someone else can comment.
Many monasteries in Thailand (including the one that our local Wat is a branch of) and elsewhere teach the Mahasi approach. The primary teachers I've had here have spent time in various Burmese and other Mahasi-oriented centres. However, they have no specific affiliation. A number of Western lay teachers (such as Joseph Goldstein, Steve Armstrong, Patrick Kearney, to mention a few whose talks I've listened to on the Internet) have a strong basis in this approach.
I'm ambivalent about "approved" styles. There are some organisations that are very organised in terms of a particular style, such as the Goenka centres. Others, such as the Ajahn Chah group of affiliated monasteries, teach a wide variety of styles that seems to depend on the particular background of the teachers. Personally, I'm more interested in access to competent teachers of the Buddha-Dhamma, than purity in some particular modern approach. However, I would always recommend sticking with a particular approach for quite some time (months at least), not randomly switching.