For a simple, efficient power tool approach I would consider a plunge router using a known straight edge as a guide. Using a straight bit you could cut in where you want and exit just the same. Use the guide to keep your cut true.
Image from PopularWoodworking.com
If you think of these grooves as flutes there are jigs that can help with the process as well.
Image from Rockler.com
There is a way to still use the table saw
If you only have the table saw or want a way for that too work there is something you could do. Take a board that is of the right height and make it 15 inches long. Cut your groove from end to end using the tips you found. Then what you do is add the stops (appear to be about an inch) and glue them / screw them onto the ends. In case I lost a reader here the picture belong can help show where the how the 3 pieces would be attached.
You can do that all with a table saw. If you use different wood then you can create interesting accents on the drawers.
I love traditional tools and techniques so if I had the time I would be using chisels that are of the same width as you groove. To clean out the bottom of the groove you could use a router plane or poor man's router. You would use a knife wall to cut your edges and use those as a guide to keep it straight as you work.
The techniques I talk about in the second part of this answer would be applicable here as well.
For your project you would have an acceptable margin for error which would allow to make a few mistakes. If these are for drawers then errors could easily be hidden.