How to make a straight groove easily?
This can be done most easily with power tools as discussed in previous answers.

However, assuming that you can create a good guide using a metal straightedge clamped to the workpiece, and you can clamp the workpiece to a heavy table, and you have a long hand saw, you can do either of the following:

Saw/Drill Approach


Put your fence together to create a rigid straightedge aligned with one side of your groove.  Now saw down 15 mm while pressing your saw up against your fence. Mark your saw at 15mm of depth so you can see how deep you've gone.
Move the fence to the other side of the groove, and repeat.
Mark a 3.5 mm drill bit at 15mm of depth. Mark the centerline of of your groove with a pencil.
Drill multiple vertical holes along your groove to a depth of 15 mm.  This is much easier with a drill press; you will drill 40-50 holes per plank.
Using a small chisel, break out the remaining pieces, and try to make the bottom reasonably flat.


This will work, but is tedious, and requires your initial 2 cuts to be very accurate.  Most people would simply find someone who had the correct tool, and who could do this in 2 minutes with it.