Acuña was born in Avellaneda, Buenos Aires, Argentina, emigrating to Paraguay (his father's origin country)[2] at a young age and starting playing for Club Nacional in 1989.

After five years, he decided to acquire the country's citizenship to play for the national team, eventually becoming the second most capped player in Paraguay's history, second only to Carlos Gamarra.

[5] Consequently, Acuña attracted attention from Deportivo de La Coruña, which bought the player for five years and €11 million even though he was still due a five-match suspension from the previous season, where Zaragoza was relegated.

[13][14] He played all the matches for the national team in all three editions, and was the first Paraguayan to be sent off in a World Cup when he elbowed Germany's Michael Ballack in the last minute of the 0–1 round-of-16 loss on 15 June 2002 (for a second bookable offense).

[15] In the build-up to the 2006 World Cup, Acuña made headlines in Sweden after reportedly having asked a FIFA employee to phone and try to arrange a date with a female photographer.