The Phoenicians certainly tried on a few occasions, and according to Herodotus they claimed to have succeeded at least once. He recorded that Phoenicians sailed from the red sea around Africa in about 600BC, and returned through the straights of Gibraltar. Sadly his account is very short on details and I haven't been able to find any other accounts of what they found or saw on the voyage. Herodotus does note that they found the sun was on their other side on the way home which some people take as evidence the story is in fact true (although he thought it was evidence the story was fabricated).

http://www.livius.org/sources/content/herodotus/herodotus-on-the-first-circumnavigation-of-africa/