Empirically, it is not possible to directly connect Linux (and therefore also not MMA under Linux) to a MS Access data source in an acceptable amount of time. 
An alternative solution is to migrate the .accdb -> MySQL DB, which then can be queried platform independently. How to connect to this DB is described in the MMA documentation (works out of the box). The migration step is, according to MySQL whitepapers, easily possible - I will update this post if this is the case. MS Access can still be used as a convenient frontend to the MySQL DB.

Network database access - How do I connect to a remote database?
The following function is supposed to return information relating to a particular meeting, stored in a meeting table:

CREATE TABLE  "MEETING" 
   (    "MEETING_ID" NUMBER(10,0) NOT NULL ENABLE, 
    "TIME" TIMESTAMP (4) NOT NULL ENABLE, 
    "LOCATION" VARCHAR2(40), 
    "MAP_HREF" VARCHAR2(140), 
    "FK_INTEREST_ID" CHAR(4) NOT NULL ENABLE, 
    "ADDITIONAL_INFO" CLOB, 
    "PASSED" NUMBER(1,0), 
    "TITLE" VARCHAR2(20), 
     CONSTRAINT "MEETING_PK" PRIMARY KEY ("MEETING_ID") ENABLE
   ) ;


The code compiles just fine, and runs fine as well.

However, if the meeting exists, only null is returned. If the meeting doesn't exist the exception prints 'UNKNOWN APPOINTMENT' correctly.

CREATE OR REPLACE FUNCTION GetMeeting
                    (meetingnumber MEETING.MEETING_ID%TYPE)
RETURN VARCHAR
IS
    CURSOR current_meeting(meetingnumber MEETING.MEETING_ID%TYPE)
    IS
        SELECT TITLE
        FROM MEETING
        WHERE MEETING_ID = meetingnumber;

    r_meeting current_meeting%ROWTYPE;
BEGIN
    OPEN current_meeting(meetingnumber);

    FETCH current_meeting INTO r_meeting;

    IF current_meeting%NOTFOUND THEN
        r_meeting.TITLE := 'UNKNOWN APPOINTMENT';
    END IF;

    CLOSE current_meeting;

    RETURN r_meeting.TITLE;
END;


SELECT GetMeeting (27) appointment
FROM MEETING;

