Getting to Furano
The travel time to Furano from New Chitose/Sapporo Airport is about the same as getting to Niseko from the same airport just in the opposite direction. The two main ways to get to Furano would be by bus or train.
Buses To Furano
There are several companies offering bus services between New Chitose/Sapporo Airport, Asahikawa Airport and Furano and are probably the easiest and least stressful way to get to resorts in Hokkaido
Here are some links to bus services to Furano:White Liner -from New Chitose Airport or Asahikawa Airport
Hokkaido Resort Liner
Trains To Furano
The trains in Japan are very reliable and a great way to get around and see the country. Getting to Furano by train from New Chitose Airport is a little more complicated than getting the Bus as you need to change at Sapporo Station for the train bound for Asahikawa then change again at Takikawa for the train to Furano, but can be a good alternative if your flight arrives at an inconvenient time for catching a bus. Direct Trains from Asahikawa to Furano are available and run frequently.
Here is the link for trains in hokkaido:JR Hokkaido
Here is a link to the Furano Tourist Board's information on getting to Furano.