Warning: Don't read the plots/synopses of the wiki links (I posted those just to give a quick impression & overview)
Butch Cassidy & the Sundance Kid - http://en.wikipedia.org/wiki/Butch_Cassidy_and_the_Sundance_Kid
1969, best buddy movie ever & one of my all-time favourites; melancholic & funny; brilliant screenplay by William Goldman, wonderful music by Burt Bacharach; everyone loves this movie, even Western haters (as this is not really a Western)
Charley Varrick - http://en.wikipedia.org/wiki/Charley_Varrick
1973, best "serious" Walter Matthau movie and one of those many great 60s/70s US road/crime/action thrillers, director: Don Siegel, music: Lalo Schifrin
Coogan's Bluff - http://en.wikipedia.org/wiki/Coogan%27s_Bluff_%28film%29
1968, another one of Don Siegel's fine 60s/70s movies
Thunderbolt & Lightfoot - http://en.wikipedia.org/wiki/Thunderbolt_and_Lightfoot
1974, director: Michael Cimino; starring Eastwood; better and more sophisticated than Siegel's Dirty Harry imo
All the President's Men - http://en.wikipedia.org/wiki/All_the_President%27s_Men_%28film%29
1976, best political thriller ever (despite "Z", "I as in Icarus" and others); director: Alan J. Pakula, screenplay: William Goldman
The Princess Bride - http://en.wikipedia.org/wiki/The_Princess_Bride_%28film%29
1987, a huge flop then, but a very funny & underrated fantasy movie mainly due to the excellent screenplay by William Goldman; Goldman is an outstanding dialogue writer
Once Upon a Time in America - http://en.wikipedia.org/wiki/Once_Upon_a_Time_in_America
1984, Sergio Leone's swan song and one of the best modern* US gangster movies not coming from Coppola or Scorsese; fantastic set design, over-the-top Morricone music (as always), brilliant performances by DeNiro & Woods (*modern in terms of not from the 30s to 50s)
The Getaway - http://en.wikipedia.org/wiki/The_Getaway_(1972_film)
1972, director: Sam Peckinpah, music: Quincy Jones; did I already mention the tradition of great 60s/70s US road/crime/action thrillers? Well, this is the best one along with the next one I recommend:
Bullitt - http://en.wikipedia.org/wiki/Bullitt
1968, director Peter Yates' best movie by far; contains the best car chase scenes ever done (and one of the most famous cutting errors in movie history: the green VW beetle)
The Thomas Crown Affair - http://en.wikipedia.org/wiki/The_Thomas_Crown_Affair_%281968_film%29
1968, director Norman Jewison, sexy, thriling, clever, cool heist movie; 60s galore; best Steve McQueen movie along with the two mentioned above
The French Connection - http://en.wikipedia.org/wiki/The_French_Connection_%28film%29
1971, director William Friedkin; Gene Hackman's performance as Jimmy 'Popeye' Doyle is sensational; contains the 2nd best car chase ever done; if you like this don't miss the nearly as brilliant 1975 successor French Connection II directed by John Frankenheimer (he also directed the excellent 1998 movie Ronin already recommend above by Dogs Rule); the 60s/70s US thrillers/crime movies are waaay more sophisticated and "deep" than anything comparable created today; and those directors really knew how to shoot on location (another gift that seems to be lost nowadays).
I could go on forever and recommend lots of movies from other genres/decades as well, but I don't want to bore you to death.