Dennehy was a talented underage player in both soccer and Gaelic football. In the latter sport, he won the 2004 Munster Minor Football Championship with the Kerry minor team and also played with the Austin Stacks club in Tralee. He attended Irish language primary and secondary schools and is fluent in the Irish language.
Dennehy began his soccer playing career at local club Tralee Dynamos until the club disbanded and he joined Kingdom Boys. There, Dennehy was called into the Shelbourne team. After trials with Nottingham Forest, Aston Villa and Southampton, Dennehy was brought to Sunderland by Mick McCarthy after the midfielder impressed during a trial at the Stadium of Light. He signed for the Black Cats in January 2005.
Dennehy did not make a first-team appearance for Sunderland, but was a reserve and youth player at the club, and was given the squad number 34 for the 2006–07 season.
On 22 November 2007, Dennehy made the move from Wearside to Lancashire, joining Accrington Stanley on a one-month deal. There, he made his first ever league appearance on 5 December against Rochdale. After a further 6 appearances in June 2008, Dennehy was released by Sunderland.
On 27 August 2008, Dennehy signed for League of Ireland side Derry City. After just a few months, however, he opted to uproot and join Cork City, signing a two-year contract with the club on 27 January 2009.
On 25 January 2010, Dennehy moved from Cork City to Shamrock Rovers. He scored on his debut in a preseason friendly at Longford Town. He scored his first league goal for the Hoops on 9 April.
In 2010 and 2011 he won the League of Ireland championship with Shamrock Rovers. He also won the Setanta Cup in 2011 and played in the Europa League group stages netting against PAOK in November 2011:
In 2011, he was club top scorer (in all competitions) with 16 goals.
On 13 November 2013 Dennehy was released by Shamrock Rovers and 13 days later he rejoined Cork City and would be part of John Caulfield's squad for the 2014 season alongside his brother Darren Dennehy. After 2 successive seasons as runners up in the league to Dundalk, as well as 2015 FAI Cup runners up to the same opposition, it was announced that Dennehy would not remain at Cork for the 2016 season.
On 9 December 2015, Dennehy signed for Dublin club St Patrick's Athletic, along with younger brother Darren. Dennehy scored on his debut for Pats on 20 February 2016 in a 3–1 loss away to Bray Wanderers in the Leinster Senior Cup.
In January 2018, he was one of the first signings announced by new Limerick FC manager Tommy Barrett.
Dennehy has played at U17, U18, U19, U21 and U23 levels for the Republic of Ireland national football team making his U21 debut in February 2007. He made his U23 debut in May 2010. He was part of the winning team in the Under-19 International La Manga Tournament in Spain in 2006.