I had one wisdom tooth pulled about a year ago with local anaesthetic (novacain) to save money. Also, my tooth was fully emerged and not impacted and it was easily removed. If yours are impacted and/or grown crooked the removal might include a more "intense" procedure and you may not feel up to riding home.
They gave me Vicodin to take (once I got home) as well as some other drugs. In my case I didn't need the Vicodin but just used the Ibuprofen. I drove the car that day. They wouldn't have allowed me to drive if I had general anaesthesia. They didn't want me to do much of anything for the rest of the day and I had a very straightforward removal.
It is true that you want to carefully follow their advice on care of the holes. I had to keep pressure on some gauze over the hole for a while afterwards to stop bleeding and I think to help a clot form in the hole. As another commenter said, you want a blood clot to form in the holes and not get what they call dry socket -- that is bad news. I doubt I could have kept pressure on the gauze properly if I had been biking.
I would err on the side of caution. I would also ask the doctor what they think.
A nice perk where I went is they send home a pint of Haagen Daz with the patient to "ease the discomfort."
