Yes, there is a difference.
Whenever you have a conductor moving across mag field lines, there is a current induced in the conductor. In a transformer, you have varying field line densities moving across the conductive material of the core, back and forth as the AC in the primary switches back and forth, which is the same thing--relative motion of field lines (or densities) and conductors. If you are moving a flat plate or a chunk of metal at 90 degrees to the field lines, or vice versa, there will be "eddy currents" induced in the metal that circle around the field lines in closed loops. The resistance of the metal causes these eddy currents to dissipate power in the metal---heating it up, and also these eddy currents make their own fields that oppose the relative motion of the lines and conductor.
So the laminations are oriented as to interrupt the eddy currents and reduce power losses to core heating. If you can imagine the direction of the field lines, the laminations should be parallel to them, because the eddy currents will try to be in circles around the field lines, and thus will encounter a stack of separate, insulated (by varnish or oxide coating) plates instead of a conductive plane to swirl around in freely.

If you are making a DC electromagnet, you may not need laminations at all, but all magnets are most effective if your design provides a complete, high-permeability "circuit" for the field lines, just like an electric circuit. This is the reason for "horseshoe" magnets or magnets that have pole pieces close together: to make a tight magnetic circuit so as to concentrate the field lines where they can be most useful, rather than spreading out into space.

I laugh at Bedini's "monopole" designs, and all other designs that only use one end of the magnet, letting the other end drag past metal bits or just spread out into space.

Oops...sorry, /rant.

Hi,

I edited a picture taken randomly from the web how the laminations are oriented in a transformer, see the attached picture. In Figure 1 I show only 3 laminations, just for clarity but there are many more of them next to each other of course.  Then I cut the core and rotated it to be about like your would-be electromagnet core, see Figure 2. I hope it shows now how your laminations should be in the electromagnet.
The magnetic flux should be guided in thin cross-sectional area: the thinner the lamination the lower the loss by eddy currents. So here comes as TinselKoala explained:
"If you can imagine the direction of the field lines, the laminations should be parallel to them, because the eddy currents will try to be in circles around the field lines, and thus will encounter a stack of separate, insulated (by varnish or oxide coating) plates instead of a conductive plane to swirl around in freely."

In the picture the dashed lines show the flux and the laminations are parallel with the flux lines. I assume your near C shaped electromagnet core will have a nearly closed flux path when the yoke part (that surely will have an I shape area) approaches the ends of the C core.

Gyula

I'm not clear enough to figure out which is better, the parallel or the perpendicular.

Thanks for the comments.

Yes, that is exactly my problem. Cutting out the frame like shapes is prohibitively expensive, but rapping strips around a wooden form to create the shape is doable.

Looking at TinselKoala  comment, there is not that much difference between the two as far as one being superior to the other eddy wise, if I am reading it correctly.

If the consensus is that there is little difference between the two, I will proceed with the strips wrapped around a wooden form.