The terminology you are looking for is a "RAID level migration".
According to this, it's possible. I haven't done it, but the procedure looks like you should add the new drive as a hotspare to the existing array, then use mdadm to update the raid level and the number of raid devices..
You'll need a recent mdadm to do this: mdadm-2.6.9 (eg, centos 5.x) doesn't seem to support it, but mdadm-3.1.4 (eg ubuntu 11.10) does:
Grow Grow (or shrink) an array, or otherwise reshape it in some way. Currently supported growth options including changing the active size of component devices and
changing the number of active devices in RAID levels 1/4/5/6, changing the RAID level between 1, 5, and 6, changing the chunk size and layout for RAID5 and
RAID5, as well as adding or removing a write-intent bitmap.
EG, add a new hotspare device, /dev/sdg, to the RAID5 array first:
$ sudo mdadm --manage /dev/md/md0 --add /dev/sdg
Then convert into a RAID6 array and make it rebuild to a clean state. The --raid-devices 4 tells you how many drives you have in total in the new array.
$ sudo mdadm --grow /dev/md/md0 --raid-devices 4 --level 6
I have no idea how quick this will be though. In my experience with doing raid level migrations on hardware RAID controllers, it's been quicker to create the new array from scratch and recover your backup to it.