Both the horizontal and vertical slats will be the same shape if your slots are halfway deep.
If you have a table saw you could use it to cut slots halfway deep in a larger board, e.g. imagine cutting horizontal slots in a 2x4.
Then once you have cut all the slots use the table saw to slice your slotted board into thin slats. Use a zero clearance insert or a sled. Watch out for kickback (or just aim your table saw in the direction of your pile of slats, heh).
This will minimize the number of cuts you have to make and also lets you cut your slat widths to whatever you want to match your slot widths. If your table saw blade is too narrow use a dado set or make multiple passes per slot.
You can keep the slots evenly spaced with a jig of some sort with piece of something on it that fits in the slot and is the size of one of your grid cells away from the blade. Cut first slot, then place it over the tongue piece and cut the next, and so on to get quick evenly spaced slots.
Alternately, like rob's answer says, if you already have slats you can clamp or double sided tape them together to cut the slots with a table saw in all of them at once, as with cutting the pre-sliced board above. The same slot spacing jig can be used if you do it this way, too.
With appropriate clamping the latter (cut all slats at once) could be done with a buzzsaw, you'd have to be creative if you wanted a spacing jig. The former (slot board then slice slats off) would be a bit trickier.
By the way I would err on the side of cutting your slots slightly too deep rather than too shallow. Slightly too deep you can still create a nice flush grid and glue will still hold fine. Slightly too shallow and you won't be able to stick your horizontals and verticals together without recutting. Make test cuts first!
By the way if you do cut your own slats and you are leaving the grid bare, another thing to consider would be gluing a thin piece of some other kind of wood to the top and bottom of your board before slotting and slicing, if you want a nice contrasting trim on the edge of the slats.