Support

Did previous backups include DOCman's files? If they didn't the issue is unrelated with the upgrade.
Are your DOCman files stored in a subdirectory inside or outside your web site's root? If they are outside the web root you need to use the Off-Site Inclusion feature of Akeeba Backup. if you don't they will not be backed up as Akeeba Backup only backs up the files and directories stored under your site's root. It can't and won't read third party components' database tables and config files to deduct where additional files may or may not be stored.
Do you get any warning about unreadable files and/or folders during the backup? If you do, that's your problem.
Do you have any exclusions set up? If you do, you may have been excluding the DOCman files which is why they were not backed up.

If you had attached the log file as requested I would have answered the last two questions myself.

What is the name of the directory where your DOCman files are stored?

According to the log file these files are backed up. Look for the log entries starting at 120731 18:28:20 and ending at 120731 18:28:25.

Have you tried extracting the file locally, e.g. using Akeeba eXtract Wizard, and check if the dmdocuments folder is there? Please note that if you are restoring to a different site / directory / subdomain / domain than the initial site you will have to update DOCman's configuration so that it looks at the correct directory for its files. If I recall correctly, DOCman uses an absolute file path.

Hm, I wouldn't expect them to say any good words about Akeeba Backup or yours truly :D I've had a fight with the owner of JoomlaTools for a different reason (because I publicly discussed the deficiencies of the framework they were trying to get other developers to use). He won't talk to me and he has his lackeys say crap about me whenever they get a chance. They are most likely going to tell you that you should not use my software and that I don't know how to write code. At least that's what their lackeys were writing on my blog comments last year.

But anyway, the problem has nothing to do with Akeeba Backup –or any other backup software– in itself. When you are moving your site between different places you have to update the configuration of all software which stores absolute paths and/or URLs including DOCman, PhocaDownloads, Akeeba Release System or even e-commerce components like VirtueMart. Akeeba Backup doesn't try to reconfigure third party software (not even those written by me – when I make a rule of not adding a specific type of third party software support I don't conveniently exclude my own software); it only reconfigures Joomla! itself.