We design the Piwik Analytics platform in a way that all the data is yours and can be easily exported. We offer many APIs which let you get all of your data, and one API in particular which lets you get all of the RAW data.
The API you want to use is Live.getLastVisitsDetails and is listed here: developer.piwik.org/api-reference/reporting-api#Live
You can use this API to download the data for a given website (or all websites), for given day / week / month / year (or all-time), and you can even specify a Custom Segment to filter out the data and return only the users that match a particular criteria.
If there is any data that you cannot export via our APIs, please let us know. It is our mission to help you keep full control of your data!