I do it quite a bit. Sling the rifle over my shoulder. A crossways rack would hang up on the overgrown trails.
I've always been fortunate enough to get game near driveable logging roads, so I bike in and either spot something right from the bike or stash the bike and walk in. One drawback is when cycling on a small old logging road or trail is sometimes you come around a corner or over a small rise and surprise an animal so you have get off the bike, to unsling the rifle and load it.
Meanwhile animal takes off.
On the other hand, deer and moose are unaccustomed to a guy on a bike and I've had them just stand there puzzled at this curious sight.
And you see more game. I ran into two guys in Region 7 one time in a big noisy diesel Ford, stop and say "Jeez, it ain't like it used to be, I guess the moon ain't right, nothin' moving around, too many hunters, there's wolves here now, the Indians ... blah, blah, blah..." the usual cliches and I responded with :
'Well I saw a doe and two fawns, another single doe, a spike buck, a wolf and a cow moose this morning."
Plus it runs on beer and taco chips.