MediaMonkey allows you to change the database's location by editing
the MediaMonkey .ini file. 

To change the location:
Close MediaMonkey (so that it doesn't overwrite any of the changes)
Back up the .ini file. If you're not sure where to find it, see:
http://www.mediamonkey.com/support/index.php?_m=knowledgebase&_a=viewarticle&kbarticleid=17 
Move the database from the default location to a new location
Add
the following line to the [System] section of the .ini file:
DBName=full pathfilename.DB (for example DBName=C:\Users\Alex\Documents\Databases\music.DB)
Restart MediaMonkey

Alternatively, install MediaMonkey with the "Portable Installation" option
checked during installation. The database will then be saved in the folder
\MediaMonkey\Portable\