Restore MySQL from files (without dump)

Is it possible to duplicate a MySQL database from their files? [I know mysqldump would be the best method to duplicate a db, but that's not possible as all we have is the backed up files from the mysql folder].

We have the ibdata1 file, the ib_log* files, and the full directories for the three db's we want to restore from the backed up files (folders seem to contain all needed frm and par files). Obviously already tried just to copy all the files into /var/lib/mysql and though it appeared the structure was intact attempts to access the data were unsuccessful.

i.e. show databases will show the db's, use dbname works, and show tables properly displays the tables, but when trying to access the data from [any] table (via query) it says ERROR 1146 (42S02): Table 'dbname.dbtable' doesn't exist - despite mysql having happily showed us the table does exist when we did show tables.

Should also point out the service was stopped prior to copying files and all files chown'ed to have mysql as owner and then the service restarted prior to attempting to access the data.