Most motherboards will run OK with a regular ATX PSU (or with the adaptor cable), although there are some that will refuse to boot at all.
Either way, in virtually all cases mobo manufacturers strongly recommend that you use a proper EPS12 compliant PSU, and there's no way to guarantee stability without it. If you're buying a new board/PSU, I'd either get an EPS12 compliant PSU, or stick with a regular ATX mobo.