Usually during hurricane season. While officially its June-November, August and September seem to be the most discounted months and from past years it seems those months tend to have the most hurricanes.
Celebrity Century 1/09
Carnival Destiny 10/07
Carnival Destiny 9/06
Monarch of the Seas 3/06
Carnival Inspiration 8/05
Carnival Miracle 4/05
Carnival Inspiration 4/04
Sovereign of the Seas 4/03
if you are going to alaska the cheapest months are may and sept. you can get some really low prices then compared to the summer months. and the kids are in school so the ship is not over run with them.
The wife and I sail in Sept. Usually the 3'rd or 4'th week. Thats about as low as they get.
Keep in mind, your at the mercy of the weather. You could have sunny blue skies, smooth sailing, and 100 degrees, or tropical storms, hurricanes, and rough seas ! Some ports of calls may be cancelled due to bad weather. Thats the risk you take.
So far, the wife and I have been lucky.