Grove was drafted in the second round (45th overall) of the 2004 NFL Draft by the Oakland Raiders, where he played from 2004 to 2008. After being the reserve center behind Adam Treu as a rookie, Grove split time with him as the starting center during the 2005 NFL season, in which the Raiders had a 4-12 won-lost record, then won the job the following year, the 7th starting center in Raider history, starting all 16 games, when Treu became his replacement for a 2-14 team. But in the 2007 NFL season, Jeremy Newberry became the starting center, as Grove started only two games for the 4-12 team. He got his job back in the 2008 NFL season, starting in 12 games, though the team's fortune stayed poor with a 5-11 record. His Oakland tenure was hampered by injuries, and he only played in more than 10 games twice during his six seasons in Oakland.
On March 2, 2009, Grove signed a five-year, $29 million ($14.5 million guaranteed) free agent contract with the Miami Dolphins. The previous starting center for the Dolphins, Samson Satele, was traded to Grove's old team, the Raiders. Grove suffered through another injury plagued year, playing in 12 games, starting 10 of them. In the 2010 preseason, he suffered from nagging knee and shoulder injuries, and was cut by the Dolphins less than a year after signing his lucrative free agent contract in favor of Joe Berger. He reportedly worked out for the Raiders, Baltimore Ravens, Detroit Lions, and New England Patriots, but did not sign with another team, and is still a free agent.