I assume, for the purposes of this post, that you are referring to Roman Catholics who are interested in transcending their traditional role as laymen and aspire to become semi-monastic.
The practice of Orthodoxy, IMO, is inherently more monastic than the practice of Roman Catholicism. It imposes a high degree of asceticism on the laity (monasticism-lite). Consequently, there is not as stark a contrast between the layman and the priest or monk. We are all called upon to be saints. The Orthodox Church is also a fairly "shallow" organization, as organizational structures go: less hierarchy. Therefore, less feeling of isolation from the leaders of the church. Plus, it has been using the same liturgy for centuries--a liturgy composed largely of quotes from the Bible, and as nourishing now as it was when it was originally created. I don't need to join a lay group to find Orthodox who are as serious about religion as I am. Other than the kinds of groups already formed to address certain issues and constituencies--as podkarpatska notes--what need is there? I'm in the only "lay group" I'll ever need.