From 1990-1994, Grigore was a member of the U-18 Romanian national chess team. His successful performances in these years helped the team earn the title of national champions consecutively from 1990 till 1993. During the U-18 national championships, he emerged as an individual champion in 1992 and 1993. He became National Champion of Romania in 1998.
Grigore has competed in various international tournaments. He took first prize at the 1997 Kremlin Cup, followed by victories in the 1998 and 1999 edition of the Lev Gutman Memorial in Tel Aviv, Israel. He won the 2001 Trofeo Statale Biblioteke and the 2003 Eforie Open tournament, both held in Romania. Grigore also participated in the Chess Olympiads of 2002, 2004, and 2006, representing Romania.
Grigore has also made notable achievements in correspondence chess. He became an international master in correspondence chess (IMC) in 2006 and was crowned Romanian champion of correspondence chess in the same year. In 2017, he became an international grandmaster in correspondence chess (GMC), one of the highest titles awarded by the International Correspondence Chess Federation (ICCF).
Apart from being a player, Grigore is also an active organizer and promoter of chess. Since 1999, he has actively campaigned for the recognition of chess as a sport in Romania and has organized various editions of the "Mica Meciul Secolului" (Little Match of the Century), a national competition for young players. He is also president of the Sector 6 Chess Club in Bucharest and a member of the board of the Romanian Chess Federation.