How wide and deep does this shallow groove need to be; does the bottom need to be flat??
Your profile shows that you have a table saw. Have a dado set? If you do have a dado set for your table saw, cut the groove with a dado cutter and feather boards to hold the work down on the table saw top and against the fence.
Or, if you have a rip blade that will cut a flat kerf bottom, use it to cut the groove and adjust the fence to the needed width. Again, use feather boards.
In my opinion and experience, there is far too much room for error/drift with a 36" long board and a hand held router, even with an edge guide. If you want to utilize a router, fabricate a guide with two runners to keep the router from wandering.
Last edited by Ray Newman; 11-12-2016 at 03:50 PM.