Same with entering Australia. About a month ago, I had to unpack a large bag because the screening agent had spotted "boots" in there -- my cycling shoes with cleats -- at the bottom of the bag. He was determined that the cleats were clods of earth... and looked quite stunned and confused when he saw the brass Atac cleats. He was very apologetic.
You will get questions about your bicycles, and where they have been and if they are clean. Likely you will be asked to open the box either before or after it has been through the xray screening.
You also will be required to fill out an immigration card on the inbound plane that you have to present to customs and immigration. There are questions relating to soil and farms.
Unlike many others, I maintain that honesty is the absolute best policy, from declaring your bike at check-in to telling customs and immigration where you have been. Even if it means being taken aside for a closer inspection. In the case above, I told the guy it was all OK because I worked in the rural industry, and was happy he was checking.
Customs and immigration in Australian, obviously NZ, the US, Canada and the UK are no longer mere formalities and you should build an extra hour or two into your itineraries for connections to take account of this.
On my way out of Australia last time, the luggage handling system for the entire international airport broke down, and everything had to be moved to planes by hand. My flight was two hours late taking off, including an hour sitting on the aircraft.
A woman (as it turned out, and Aussie) turned on a real hissyfit at the Air Canada counter when she was told she was too late to board the connecting flight. She had just arrived from Sydney, and her baggage took an extra 30 minutes to come out at LAX (not connected to the system breakdown in Australia). Her connections were just too fine, and I gather she was in deep pooh for missing business meetings in Vancouver.
This is probably not all relevant to avatarworf and their plans, but threads like this, I think, are useful for chatting about these issues so others planning similar trips are a little more familiar with what can and does happen.
I'll stop now.