Well, "easy" is a relative term, but generally speaking if the MySQL version at your present host is similar to that on DreamHost (DreamHost uses version 5), the process is relatively straight-forward.
Essentially, what you need to do is:
1) Export the data from your old host's database(s) to .SQL files.
2) Create new database(s) on DreamHost to hold the data. This is done from the DreamHost Control Panel (Goodies -> Manage MySQL).
3) Import the .SQL data that you exported from the old host database(s) into the new DreamHost database(s).
4) Modify the phpBB or other application's configuration to allow it to connect to the new "DreamHosted" database.
Depending upon the tools available at your old host, you can do the exports using PhpMyAdmin, or the shell and at DreamHost you can do the imports using either method.
Don't assume that databases are the only "catch" when migrating database-driven applications/websites though, as different versions of PHP, and how PHP is run (mod_php or PHP-CGI) can have an impact on how easily the code migrates between hosts as well as different versions of the applications themselves.
Additionally, these applications may well have their own detailed processes for migration that can include the changing of configuration files, etc.
Most applications support forums or codex's will have instruction for "moving" an installation, so you should research these for specific tips related to your phpBB and other applications.
Lots of other help is available via Googling, but if you have a specific problem, or need a "step-by-step" for a particular thing and you can't find it elsewhere, I and others here will gladly help if we can.