When I was a kid my friends and I just carried our rods in one hand while controlling the bike with the other hand. I still see people -- children and adults -- doing the same thing today, but there are fancier options available: special mounts for carrying your fishing pole on your bike hands free, and collapsible "fishing pen" rods that you can carry in a backpack.
A fishing rod not mentioned by Canfield are child fishing rods. When my daughter outgrew her short Disney Princess fishing rod it's a short enough rod that I used it for bike trips to the lake a few times. Child rods are cheap but they're also very portable.
The Shimano Pocket Fisherman inserts into the end of a mountain bike flat bar. It uses the rear STI shifter as a reel and the rear shift cable as line. The frayed cable end forms the hook. Fiendishly clever.
As a kid, I used to tie my spinning rod to the frame of my Sting-Ray with twist ties. More recently, I've carried my less-expensive fly rod into the backcountry by strapping it diagonally across my hydration pack, dangling the reel as low as I dare so the ferrules aren't too exposed at the top. For road accessible trips, I just use a Lone Peak grocery bag on my Long Haul Trucker. It breaks down to about four feet, so it's not as though it's snagging power lines. Happy Trails, Ron