You never, ever want to cut a tendon with a router. Or with anything for that matter.
Tenons are another thing.
I use my table saw to cut them. I have seen some very cool set ups using a router mounted horizontally though.

 

Some people say "tenon", I say "tendon."
--Captain Eddie

 

For that matter, my Mother used to say....
Hawaiya for Hawaii
worsh for wash
and Baskin Robinson for the ice cream place.

If Captain Eddie said he just cut a tendon, how would you know what to say? Good job? or I hope you get out of surgery soon!

 

Some tips on router morticing.

 

woodgears.ca has a lot of jigs and plans for woodworking.

 

I almost cut a tendon, ok just my fingertips, on a jointer working in a cabinet shop, not a good feeling. But I'm glad this was asked, looking for the same answer.

 

I have had a couple of router books that can be bought at places like HD, all of which have sections on different ways of mortising. The one that's lying on my floor right now is the Complete Illustrated Guide to Routers, by Lonnie Bird.

 

Hi

Many,many ways to get the job done But the best one I have found is the floating type, it's so quick and easy and ,no guess work needed..

JessEm Zip Slot Mortise Mill - YouTube

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Tendons & ligaments are body parts critical to movement and strength.

Tenons are wooden "rectangular dowels", of which there are a couple of types: (1) loose (a.k.a. "floating" as BJ mentioned above) tenons and (2) integral tenons. For a single loose tenon connection there are 3- pieces of wood (typically). For an integral tenon joint there are 2- pieces of wood, which can be thought of as male and female parts. There are advantages to either methodology - usually it is maker's preference.
Personally, I use a router to make mortises and I use a table saw to make tenons. If I am making integral tenons - I use a tenoning jig - which is a purpose-built table saw sled. If I am making loose tenons - I use the table saw for tenon width and thickness and I use a miter saw to cut the length (cut the length last). Tenons should be slightly shorter than the mortise depth. In a majority of cases, it is easier to make the mortise first - followed by the tenon being made. If you are mortise cutting the end grain of wood, be very careful as that is the hardest surface of a board to penetrate with your router bit. BE SAFE!