The most played board games are old; some are even ancient (#2 most played, Checkers; #3 Chess; #13 most played, Backgammon). The only game to make the list from this century—and just making the list, at #22—is Ticket to Ride. 

Here are the top games sorted chronologically from oldest to newest:

DateGame% Played Last YearRank
~3000 BCEBackgammon  8%13
~3000 BCECheckers 25%2
~600 CEChess 24%3
1630Cribbage  4%18
1860The Game of Life  9%11
1929Sorry! 12%9
1931Battleship 17%6
1935Monopoly 37%1
1938Scrabble 17%5
1943Chutes and Ladders  8%12
1948Candyland 19%4
1949Clue 12%8
1956Yahtzee 17%7
1957Risk  5%17
1965Trouble  9%10
1974Sequence  3%21
1977Rummikub  3%19
1979Trivial Pursuit  8%14
1985Pictionary  6%15
1995Catan  5%16
1998Cranium  3%20
2004Ticket to Ride  2%22

The median year is 1949 and most “modern” boardgames date from 1929 (Sorry!) to 1965 (Trouble). The 1930s, 1940s, and 1970s are the only decades to have three games among the most played.