Some probably also used Vox organs as well. But I agree with Russ, if I was playing 60s covers I'd definately consider the Nord Electro 2 http://clavia.com/main.asp?tm=Products&clpm=Nord_Electro_2&clnem=Information

Good Hammond organ sounds as well as Fender Rhodes and Wurlitzer piano ( both used a lot in the 60s ) and also good Clavinet emulation. The Electro 2 comes as either a light portable keyboard or as a rack module.
Plus they really aren't too expensive compared with many other keyboards. Check eBay for deals on it.