Start by searching for a flight like you usually would at jetblue.com, but before clicking the button to search for flights, check the box just above that says Use TrueBlue points. The search results you see will include any available JetBlue flights and those from partner airlines, like Qatar and Hawaiian.
You can redeem 70,000 TrueBlue points to book Qatar Qsuites between New York and Doha. That same flight in economy would cost 34,000 TrueBlue points. With the Chase transfer bonus, that means you could fly Qsuites for 56,000 Ultimate Rewards and Qatar economy for about 28,000 Ultimate Rewards transferred to TrueBlue.
These rates are on par with what you would expect if you booked the same Qatar flight through British Airways Executive Club or American Airlines AAdvantage, two of its Oneworld Alliance partner programs.
If you plan to use Doha as a gateway to a more tropical destination like the Maldives, you can book Qatar business class one way from JFK to Male, Maldives, for 106,000 TrueBlue points or 58,000 TrueBlue points in economy. With the transfer bonus from Chase, those totals drop to about 85,000 for business class and 47,000 for economy.