If I understand this correctly, it's not really a new rule. They're just adapting the one here in Canada. From my understanding, the ground clearance would sound something like "Taxi Alpha across 22 for 32" Where the runway you're expecting to take off from is 32, and the one you had to cross is 22. If that's what is now being enforced in the US, then it's not so bad. It has been in Canada for at least as long as I have been flying, which is coming up on 4 years now.
Its a little more official than that. You now have to have a clearance to cross the runway. It would be "Taxi to runway 32 via Taxiway Alpha, cleared to cross runway 22". They also can not clear you to cross more than runway at a time, so if you cross more than one on your path, they would have to wait for you to cross one before clearing you to cross the other.