My main criticism is that Edmodo seemingly started as a microblogging tool, which it is good at, but teachers have started using it in place of a VLE. There's nothing wrong in that per se. However it has the problem, in common with any VLE that is not based on your network in that it takes ages to download posted work, whereas if the student uses a blog, such as Posterous the work is nearly always in a format that can be viewed immediately on screen, and the whole body of the students' work can be viewed just by scrolling down the screen..
Of course with a blog, the work becomes viewable in the public domain. This has benefits as well as downsides.Some teachers/parents/students may feel uncomfortable about having student's work being so openly on display, but is does give the kids more motivation to do a better job and they can also peer assess work, unlike on Edmodo or a VLE and you can also bring in e-safety issues should they arise. The other nice thing is that it can become part of an online portfolio that can help show their progression as they get older. If you want a walled garden approach one local school tried using ELGG as a private blogging platform, on its own server, but for whatever reasons they now seem to use open blogs for work submissions. If you decide to go down the blogging route Matt Baker at Pattana put together a nice set of How to videos for using Posterous.
I've had my Year 7s submit much of their work this year using blogs and Google Sites http://bit.ly/
y7sites2012 and find it far easier to check through than using Edmodo. I tried using Google Reader to monitor submissions, which many teachers use, but found that an online spreadsheet suited me better. I still use Edmodo for posting assignments, mainly because it is easy for students to download worksheets and stuff that I may want to post, but I am think of moving to a more open website solution next year, so parents and the school have more visibility of the programme of study
Hope this helps you decide what's best for you.