3D graphics are evidently appealing for cultural heritage communication: the possibility of (re)creating environments, buildings, objects, etc. raised great expectations, both with professionals and end-users. Actual applications, however, have turned out not be so successful as they had promised to be. Therefore we must rethink goals (what can 3D be used for?), requirements (which features should we try to achieve?) and design (how do we do it?). This paper deals with a specific niche, i.e. shared 3D worlds accessible by several users (represented by avatars) over the Internet; some of our considerations, however, can probably be extended to other types of environment. Our main point is that shared 3D worlds are effective not just because they offer a realistic reconstruction of something, but because they have the property of creating (if well designed) virtual presence, i.e. the feeling of "being there" with someone else, engaged in some kind of activity. We argue that, if virtual presence is what we look for, the main concern should switch from the quality of the graphics to the quality of the overall design of the application. The paper discusses the above issues on the basis of the experience gained in 3D educational projects developed by the HOC laboratory of Politecnico di Milano: SEE -- Shrine Educational Experience (in cooperation with The Israel Museum, Jerusalem), Learning@Europe (in cooperation with the International Accenture Foundation) and Stori@Lombardia (in cooperation with the Regional Government of Lombardy). These projects have so far involved more than 1400 students and teachers from all over Europe. All these projects are an evolution of Virtual Leonardo, winner of the prize as "best on-line exhibition" at Museums and the Web 1999.