There are two ways of making dowel on the router.  Using a fence and round over bit or a jig /die setup similar to the Table Saw method.

USING A ROUNDOVER BIT

When using a roundover bit, the critical part of cutting dowels on a router table is leaving square sections on the end of your workpiece (that’s why you cut it long).
This creates stability and accuracy when cutting the round overs. You can set up two stop blocks on the router table fence to help or just eyeball it.
Start the cut by pushing the stock sideways into the router bit until it’s flush against the fence. Make sure there’s around a 2” section of stock that isn’t cut ahead of the bit and behind it.

Once your workpiece is flat against the fence, push it through the bit. Stop around 2” before the end of the workpiece.
Flip the workpiece and repeat the process for the three other sides.

Measure diameter first to make sure its satisfactory before cutting off square ends.

PRO

Its simple and fast

CON
 
Bit set up against the fence is finicky. You must make sure that the arc of the cutting edge is tangent to the table surface and front of fence. It is important to make sure you have a good quality round over bit. The other drawback is making sure you cut exactly square stock to match the finished dowel diameter that you want. If you want 1/4" dowel, your ability to achieve this is making sure you cut as accurate as you can a 1/4" square stock. This isn't as painful as the chisel setup but some.


USING A JIG OR DIE SETUP

This is a jig or die setup. Here I made it out of a piece of 1"x 2" x 6" piece of maple. Its similar to the Tablesaw Method.
Note: The two holes are mounting holes that mount the die to a piece of plywood that can be positioned and clamped into place on your router table.

 

You drill an input and output hole and another vertical hole that is tangent to the output hole. The vertical hole is where the router bit goes.
Lets say you want a finished 1/4" dowel. You drill an 1/4" dia output hole from one side to the center of the die as shown below.
The input hole will be 1.4142 x 1/4" (the diameter of the output hole) = 0.354 . You can round this off to 0.375 (3/8") dia. if you like or keep it tight at 11/32" dia. You chuck up a 3/8 x3/8 piece of stock in your drill, turn the router on and push through.

PRO

Beautifully and smoothly cuts dowel. Very fast

CON

As mentioned with some other methods it is finicky to setup. If you want really accurate dowels then it takes time to setup for perfection. Good for mass making of dowels but questionable for the onesie or twosie.