I realize this is an old post - but I didn't see a suggestion that I think is important:
* the 32 bit install of Office is widely recommended and still most common
* the 64 bit install of Power BI is most common
I think the idea of altering either to not be the most practical approach to this dilemma.
An alternative, if it is at all feasible, is to export the Access tables out to excel. They can then be brought into Power BI without a problem.
A side benefit to this is that, if one knows the Access data, a simple query can define which data gets exported - and so you can perform the first step of Power BI work as part of this as well by not exporting unneeded fields and also delimiting the range of records.
I had the same issue (see error message below)... so I uninstalled my 32-bit version of office... and installed 64-bit version... restarted 2X... and am still getting the same error... i.e., unable to Load the Access database tables into Power BI. Hope to can help. Thanks.
Did you delete the prior connections and make brand new ones? Also sanity check your new Access install by opening it and looking at the tables. You might copy a duplicate of the Access file and change its name slightly....and then attempt to import the tables of this new file.