If possible, copying the data files will be the fastest option. Since you won't need to reinsert data or rebuild indexes. Just be sure to take the relevant backups just in case, which may include a
mysqldump, before you begin.
You can copy data files from both MyISAM and InnoDB between architectures providing they use the same floating-point format. It's foremost documented here for MyISAM and here for InnoDB. You might also be interested in this thread which asserts specifically that there are no expected format issues when moving between Windows and Ubuntu as a platform. It also includes some useful advice on the procedure.
In relation to Roland's answer, copying or moving InnoDB data is possible and supported. You simply have to be a bit more careful of all the relevant settings. In much the same way that InnoDB will fail to start if you change the data and log size options on whim.