The best way is probably to set up the new server, then schedule a maintenance window for turning off normal users' access to the old server. During the maintenance window, use a file copying tool to copy everything you want to keep from the old server to the new one. Once the copying process completes, optionally do a verify step (something like a recursive md5sum on both and diff'ing the results should do nicely) before replacing the old server with the new one. Decomission the old server (and particularly its storage devices) as required by applicable procedures and regulations.
Incidentally, this has nothing to do with whether you are using RAID or not. RAID is simply a convenient method of, among other things, gaining storage redundancy. As a side-effect, it tends to make whole-disk copying (through tools like dd, ddrescue etc) less useful. That latter is already not very useful if you are migrating to disks of a different size, as the partitions would remain the same size on the target media; it is most useful when a disk is experiencing trouble. The whole point of a redundant RAID configuration (that is, essentially everything except RAID 0) is generally to reduce the likelihood of such troubles having any detrimental effect on user access to data.
And while you're at it, consider migrating to RAID 6 (or 1+0) rather than RAID 5. Single redundancy just doesn't really cut it with today's drive sizes. Some people even predict RAID 5's rapid demise, in large part because of that. You get better disk space utilization with RAID 6, but RAID 1+0 can generally give better performance.
Oh, and you are already backing up the files on that server, aren't you? Because if you aren't, then you should start there rather than worrying about RAID this-or-that. We regularly get questions here about how to recover important files that people didn't have backups of; don't become one of them. Even more so if these files aren't your personal files, but things people need to do their work.
