Moodle is an LMS you can use in your eTwinning projects. It is a free, Open Source software package designed using sound pedagogical principles, to help educators create effective online communities. You can download and use it on any computer you have handy (including webhosts). Currently there are 26,979 sites from 184 countries who have registered (you don't have to register so the number of users is way higher). Moodle is available in 72 languages.
I have used Moodle several times in order to create online courses for teachers. Actually, my 1st eTwinning online seminar for Greek teachers was developed in moodle. I know that many teachers are using moodle for their eTwinning projects and they are very satisfied of it.
Moodle’s design is informed by a social constructionist pedagogy that emphasizes interaction over content. This platform manages tracks and reports on interaction between learner and content, between learner and instructor and between learners. Moodle performs student’s registration, schedules courses and defines curricula, delivers learning material and assessments, tracks the learners’ progress and finally allows communication by chat, discussion forum, messaging and news. Apart from that, moodle offers also standard functionalities like: collaborative tools (wiki, blogs), file management, grouping of students, polls etc. I have also used this platform because it controls the type of content explored and the manner in which it is explored.

If you want to try Moodle (without installing it on your own server) as a teacher or administrator, please visit: http://demo.moodle.org. The database and files are erased and restored to a clean state every hour on the hour, so don't worry if you make a mess.