This convertion could be obtained by first converting the tables in the mdb-file into CSV-files (Comma Separated Values) using MDB Tools and then by importing the csv-files into MySQL using phpMyAdmin.
To export a given table from a mdb-file, we type:
mdb-export db.mdb customers >> customers.csv
In my case, since I got some danish characters, I need to force mdbtools to encode everything as ISO8859-1 instead of UTF-8, which is the default. This can be done by setting the environment variable MDB_ICONV to ISO8859-1
The resulting CSV-files could be imported into MySQL using phpMyAdmin.