Sounds like a job for datapump export/import.
How big are these databases and how much downtime can you afford ?
There are various methods which can be used to achieve cross platform migration. Please see article -
733205.1 - Migration Of An Oracle Database Across OS Platforms (Generic Platform)
Choose an appropriate method depending on DB size, downtime requirement etc.
There's no switch in Endian format in that case, so you have many options.
If you supply some more details, maybe we can direct into the best method. There's 7 of them mentioned in 733205.1 (as also shown above).
Things like : do you already have a procedure to install/setup a 22.214.171.124 on Linux, or is this the first of that version and that platform.
Other questions are already mentioned, especially the one on downtime.