Apparently, there were some private citizens from the USSR who were allowed into the U.S. for travel during the Cold War. But they couldn't just visit anywhere they wanted.
This map, from a post at Slate's Vault blog, shows the no-go zones, shaded in green. Some of this is quite funny — gee, guys, I wonder what you're keeping hidden out in rural Nevada? Another interesting point: Soviets could visit Kansas City, Kansas, but not Kansas City, Missouri. Which could just be a pretty good joke, on our part. The fun stuff is all on the Missouri side.
EDIT: In the original version of this post, I'd mentioned that Kansas had once been home to many, many missile silos, and speculated that this might be why so much of that state (and the Dakotas) was off-limits to Soviet travelers. But, Cold War historian Audra J. Wolfe contacted me and pointed out that there were no missile silos at the time this map was made, because there were no Intercontinental Ballistic Missiles. So why ban the Ruskies from Kansas? Wolfe isn't entirely sure. She speculated that it might have had something to do with limiting access to public lands managed by the Bureau of Indian Affairs or the Bureau of Land Management. It also could have been tied to the presence of Strategic Air Command bases in the state. And there were tons of Atomic Energy Commission-owned sites scattered all over the U.S. — it's hard to keep track of where they all were.
Of course, Wolfe also said that there wasn't always a clear logic behind the decisions about which parts of the country were made off-limits to Soviet citizens. For instance, much of our coastline was off-limits for no other reason than the fact that much of the Soviet coast was off-limits to Americans. "The main premise is 'strict reciprocity'," she wrote in a message to me. "X% of Soviet coasts are off-limits, therefore x% of US coasts are off-limits, too." So there, one might add.