Depends what you mean by 'ecumenism'. Being nice to Protestants, et al., and having official talks with them in order to teach them are good. They don't go against the true-church claim and shouldn't rankle converts' understandably strong emotions about their former faiths. That's what official Catholicism and official Orthodoxy mean by ecumenism.
Then there's the 'ecumenism' that's relativism denying the true church, which rightly gets online Orthodox' hackles up! Every now and then a Catholic posts an opinion like that. It's like the Anglicans' branch theory, where they're trying to be nice by including 'the Roman Church' (what they call Catholics: hail, Caesar!) and the Orthodox as 'branches' of the true church, but it really insults both by making light of each's teaching on the true church.
Then in this online circle you get a few, mostly converts, in Greek Catholicism who thumb their noses at post-schism Catholic defined doctrines and agree with Orthodox opinions on everything, but they don't join the Orthodox. Again, thinking they're better than either church. Basically, Stuart Koehl (their doyen, who's been preaching essentially himself for nearly 20 years online), maybe Fr. Bob Taft, an occasional Melkite such as Bp. Elias (Zoghby) of the Zoghby Initiative, and some high-turnover traffic of converts to Greek Catholicism who might start there as liturgically Orthodox and doctrinally Catholic (what Rome wants Greek Catholics to be) but get fed up with the latinizations and second-class treatment so they buy into Orthodoxy and convert.
proper titles added - MK