Disclosure: I work for Microsoft, but not in the Teams group.
The APIs to read messages and threads (message replies) from a Microsoft Teams channel are now available in preview!
Accordingly - I've updated my multi-platform, .NET Core-based Teams sample application, QuickTeams, to demonstrate how you can use these APIs. You can use QuickTeams to download all of the messages in a specific channel, or across all channels in a single Team. Feel free to use QuickTeams as-is or as a starter kit for your own needs.