Hi David,
Is there any specific reason you want RTMP streaming? If not and you're happy with progressive download (http://en.wikipedia.org/wiki/Progressive_download), you can use any server that can store files, e.g. a bucket server. If you have very long video files, i.e. longer than 20 mins, then RTMP streaming is one (expensive) option whereas, if you have the original high bit rate video files (to avoid lossy transcoding), you can edit them down into 10 minute sections and deploy them in one player as a playlist (easier to do in Moodle 1.9 than 2.x but still doable).
Also AFAIK, with RTMP streaming, HTML5 isn't really a viable option yet as support for it is still experimental.
And, progressive download is a lot more robust than RTMP streaming. If a user has a slow or intermittent internet connection, e.g. mobile broadband, a computer lab with a shared connection, or just a poor service, stuttering/buffering can be a showstopper whereas with progressive download, the whole file is being downloaded in the background and all the user has to do is pause the video for a while and then resume playback.
The two main reasons for using RTMP streaming are #1 for digital rights management, i.e. to deter users from saving local copies of your video content, and #2 to allow users to scrub through long videos without having to wait for the video file to have downloaded to that point.
For an RTMP service, you could try Bits on the Run: http://www.longtailvideo.com/bits-on-the-run/ which is particularly easy to use. Another popular option is Brightcove: http://www.brightcove.com/en/ There are many other RTMP streaming server services available and they almost all use either Flash Media Server (Java-like) or Red 5 media server (Java). Running your own media server would require a skilled, experienced developer, a dedicated Java server, and cost far more than using a 3rd party service.
I hope this helps!