Falvey was a trainee and youth player at English clubs Gillingham and Carlisle United, but began his professional career in 2004 with Cobh Ramblers in the League of Ireland, scoring on his debut vs Athlone Town. He transferred to Kilkenny City prior to the 2007 campaign for a compensation fee because he was under the age of 23, and was made club captain upon arrival. During his time playing in Ireland he attracted interest from Swindon Town, but opted to sign a contract with Otago United in New Zealand under ex-Irish international Terry Phelan, who would become his agent.
Falvey spent one year in New Zealand with Otago United, where he was club captain.
Falvey signed with American side Wilmington Hammerheads in the USL Second Division for the 2009 season. He played 15 games and scored 1 goal for the Hammerheads in 2009, earning him defensive MVP honours and been selection in the league team of the season while also helping them to the 2009 USL Second Division regular season title. After the season, he returned to New Zealand on a 6-month contract to play for YoungHeart Manawatu.
On 25 February 2010 it was announced that Falvey would be returning to USL league in America to sign a one-year contract with the Charleston Battery. Falvey was a key starter in his first season with the Battery as the club won the 2010 USL Second Division championship. Falvey signed on loan with previous New Zealand outfit YoungHeart Manawatu during the USL offseason before returning to Charleston to sign a long-term contract on December 8, 2011. Falvey was named the club's captain and became a fan favorite, with part of the club's supporters' group being nicknamed "Falvey's Army" during this time. Falvey won a second league championship with Charleston in 2012.
After leaving Charleston at the end of the 2014 season, Falvey announced he would be signing with Kerala Blasters of the Indian Premier League for the final months of his contract with the Battery. Falvey made 6 appearances for Kerala in the inaugural season of the tournament.
Falvey signed with Ottawa Fury FC of the North American Soccer League in February 2015.
Falvey played every game of the Fury's 2015 season, helping Ottawa earn its first-ever piece of silverware, the 2015 NASL Fall Championship title, and its first-ever Soccer Bowl appearance. Falvey also helped the club set an all-time NASL clean sheet record of 648 minutes and concede the fewest goals of any club that season.
On 15 January 2016, Falvey was transferred to Indy Eleven for an undisclosed fee.USL Second Division Regular Season Champions (1): 2009
2009 league team of the season
Club defender of the year.
USL Pro Champions (2): 2010 2012
USL Pro Regular Season Champions (1): 2010
USL Pro League Defender of the Year 2013
NASL Fall Championship 2015
NASL Spring Championship 2016
2009/2010 Player of the year.