I've had a Greenland paddle custom made as well as carving one myself.

One was Western Red Cedar, the other local Eastern Cedar.

 Both are lovingly cared-for with regular applications of high quality tung oil.

 Enjoy! 

I've used both Boiled Linseed Oil and Tung Oil. The Linseed is more amber in appearance than the Tung.

PRO: Both are easier on the hands than a varnish, and I find that very important and the primary reason for using an oil.

CON: You need to renew the finish often. The usual creed is once a day for a week, once a week for a month, once a month for a year (or thereafter).

If you decide to use Tung, use the REAL stuff, pure Tung oil, not something bought at Home Depot or Lowes. The stuff at those home centers is not pure Tung oil. It's a mix of oil and varnish that's OK for furniture, but not for your paddles.  

I used tung oil on my greenland paddle, worked out nice was quick and easy. I also coated the tip in epoxy for a little more wear resistance for push offs or hitting those hidden rocks.