I made Philly Cheesesteak sandwiches and the resulting meat was quite tough and hard to bite off and chew.
For the meat I used eye of round steak, cut into roughly 3in by 1/8th-1/4th in strips. I did not marinade the meat or season until salt and pepper during cooking.
To cook the meat I used a nonstick pan over medium heat for a few minutes.
Is my cut of beef to blame, my technique, or perhaps both?
My experience has shown that you need to slice the meat against the grain when preparing it for sautéing. The shortened muscle fibers make for a more tender piece of meat. After that, the quicker you can cook the meat, the better. Round steak tends to be best either quickly cooked or cooked for an extended period of time -- there's not too much of a middle ground for it.
For a detailed explanation of meat toughness, see What makes a moist steak (or roast)?
As for your specific case: I also don't know about American cut names, so there are two possibilities. Either you chose a collagen-rich meat, or you chose a cut low in collagen.
Assuming that you want fry steak in the pan and not have it become tough, you should choose meat which is low in collagen. If you didn't, you have to change the cut (or not cook it like a steak, as Rorschach120 advises).
If you have tender muscle and it becomes tough, you have overcooked it. "A few minutes" sounds too much for such thin strips. They are cooked until they first show color (browned, not just grey). If they get tough until they show color, you need to increase the temperature, and possibly ditch the nonstick pan.
Another option is to fry it as thicker steaks or oven-roast it. Then you can use a meat thermometer and achieve your preferred level of doneness. Cut afterwards and sear or torch the newly exposed surface.
The toughness in meat comes from the amount of collagen in it. In order to break down this collagen so you are left with tender flaky beef, you have to either slice it very thin so it has an easier time in the pan, slow-cook the beef in a crockpot, or pressure pressure cook it (the best way IMO). After slow-cooking or pressure cooking, stick it in a KitchenAid mixer with the paddle attachment and run it for a minute.
When you cook it in a pan (really hot pan) stir it all around for about 5 seconds and then add some Worcestershire sauce or other liquid to quench a bit of the heat and then let it heat up again and it should be done by then. Happens very fast.
Eye of round is a relatively lean steak and not quite as good for cheesesteak unless you use the first method of slicing it really thinly before cooking it. I would use chuck or rib for cheesesteak.
