Several good suggestions, but I see some confusion over 'dowel' 
and 'plug'.

A dowel is made with the grain running along the length of the piece;
you can get them three feet long.

A plug is cut into side grain of the wood; you rarely get them more than
1/2-inch or maybe 3/4-inch long.  Generally you buy a plug cutter and
make your own, although marine stores and some good woodworkers supply
places sell plugs.

A dowel is all wrong: the grain doesn't match the base wood, the screw
will be placed into end-grain which is almost useless for holding
screws.  The end grain of the dowel is almost impossible to trim
exactly flush with the surface, and even if you do any movement of the
base wood with moisture will affect the dowel the wrong way and sooner
or later it will be evident.

Sticking a dowel into a new drilled hole is no better than simply
sticking a toothpick -- which is really a small dowel -- into the hole.
A properly installed plug will match the grain, will react to moisture
changes exactly like the original wood, and be almost invisible.

Ron Ginger