This is not really an answer yet, but definitely too long for a comment.
Access databases are designed for single user access and store data, forms and code in a single file. It is possible to access such a file from a remote computer via e.g. a network drive. As there is no real database server involved, concurrent usage of the same database file with several processes/programs will only be possible within certain limits (which I don't know about).
It should be possible to access such a file from a network drive via the usual ODBC drivers, but you might need to configure those accordingly. You have mentioned that you want to access the database from a linux client: There are OBDC drivers for access databasefiles which work under linux, but I don't know if and how it is possible to use those from Mathematica, but would expect that should be possible. If the file lives on a windows server you could mount a network file with the samba client programs, but that might introduce additional restrictions on concurrent usage as there could be differences in how the file locking and other details are handled. I would be very careful when experimenting with such a setup. On the other hand I wouldn't expect insurmountable problems if only one program accesses the database file at a time.
Depending on how you are using that database converting the database is probably the best you can do, especially if you need concurrent access. As a first try you could try to use e.g. openoffice or kexi to convert the database, but of course you could also do that with Mathematica on a Windows machine. Depending on the target format there might be even better options, e.g. specialized import functions/scripts for certain database systems.
If you don't need concurrent access you could consider SQLite as a target format which like Access stores the data in plain files and also is "serverless", so using it might be simpler than setting up a database server. There is a JDBC driver for SQLite, which you should be able to use from Mathematica. There is also an undocumented more direct implementation to access a SQLite database from within Mathematica even without DatabaseLink, which you should find some notes about in the web.