It's often pointed out that the excellent Reiner Knizia card game, Lost Cities, could be played with a standard deck of cards if there were one extra suit. Thinking about the simplicity and elegance of the rules made me wonder if there might exist a traditional card game with similar mechanics but just four suits/cities? (I don't know many traditional card games, so I'm not even sure if there's a broad category that Lost Cities might fall into.)
I don't think I've ever heard of such card game. To me it's pretty clear that Lost Cities were developed and designed using standard card deck, but I don't suspect it's actually based on some truly similar traditional card game. Maybe the basic idea was to design a 2-player game with simple rules, something a little like Solitaire, but competetive, with piles shared between the players?
I heard Lost Cities being described as 2 player Solitaire with an extra portion of suspense - source. If there was more similar game to Lost Cities, maybe such descriptions would mention it.
I guess this isn't the answer you were looking for, although I can't give you any game title that would have the feel of Lost Cities. This is the interview with Reiner Knizia in which he mentions origins of Lost Cities. You also may want to take a look at this interview and this one.
You might also try checking Patience category on pagat.com