There have been a number of foreign-born NFL players, many of them well known and successful place kickers. The most successful has to be Morton Andersen, born in Copenhagen, Denmark. The all-time leading scorer in NFL history and the all-time leading scorer for two teams he played for (Saints, Falcons), Andersen became the NFL all-time leading scorer on December 16, 2006 when he passed South African born Gary Anderson (Steelers, Vikings, Titans). A week later he passed Anderson again to become the all-time leader in field goals made.
Jan Stenerud, born in Akershus, Norway, is yet another foreign-born place kicker who had a very successful NFL career. One of only three place kickers in the Hall Of Fame (George Blanda and Lou Groza are the other two), he is the only one who didn't play another position (Blanda was a quarterback, and Groza played offensive tackle).
I would like to be able add Lino Dante "Alan" Ameche to this list just to have a fairly well known non-kicker, but there seems to be some confusion as to where, and when, he was born. Some sources say he was born in Italy, while others say he was born in Kenosha, Wisconsin, where he grew up. It's also unclear what the exact date of his birth was, either March 1 or June 1, though all agree the year was 1933. The confusion appears to stem from the fact that after initially emigrating the family returned to Italy for some time before returning to the the U.S. for good in the late 1930's.
If Ameche was born in Italy it would, I believe, make him the only foreign born Heisman Trophy winner. He achieved the honor in 1954 after his final year at Wisconsin where he rushed for 3,212 yards (then the NCAA record), 25 TD's, and averaged 4.8 yards per carry in a four year college career.
Drafted by the Baltimore Colts, Ameche was named the NFL Rookie Of Year after carrying the ball 213 times for 961 yards and 9 touchdowns, all league highs that season. He was also named to the Pro Bowl, the first of four straight selections.
Ameche scored two touchdowns in the Colts 1958 Championship Game victory over the New York Giants, the second being the game winner in overtime, in what is still often referred to as "The Greatest Game Ever Played."
Pete Gogolak, born in Budapest, Hungary, yet another place kicker and the first "soccer style" kicker in the NFL, began his career in the AFL with the Buffalo Bills before signing with the New York Giants where he played for 9 seasons.