The most similar app would be FileMaker Pro as it combines database and user interface features.
Though it won't read the *.mdb files. You would need to export the data as text first, create a new FileMaker Pro database with the same fields, then import that data.
Obviously you would need to do the export on a pc (or virtual pc) where Access could run.
Or as the previous post said, just get a copy of Virtual PC and run all your pc apps including Access.