Question:

We (2 dutch) travelled from 4 october to 22 november through Iran by bike. Going through Iran we didnot had any problems with police or army. People are very helpfull and friendly. In Teheran we extended out visa within 2-3 hours. this in contrary as written in the book. We got 60 days extra without any hesitation.
Biclycle on the bus is somehow complicated. A new rule is that buses are not allowed to have luggage on the roof. So bikes must be in the luggage compartment under the bus.
But when the distance from travelling is too short. Smaller buses do the trip. Then the compartments are too small. The bikes might be on the rear chairs, Then you have to pay 5 seats extra fot this.
In Masshad, we stayed with Vali's no smoking homestay. This is still a nice place to be. Vali and his family do everything to make you happy and take care. We planned 2 nights and stayed 5 nights. A visit to Kang village is a must do. Places like Masuleh and Abyaneh village are about simallar, but more touristic.
For cyclist is going east and than down to Tabas nicer, than the standard route Teheran, Qom, Esfahan Yazd, Kerman etc. From Tabas we went to Khur. We planned 1 night to go to Garmeh. We skipped Garmeh and stayed here 5 nights. Yazd we stayed in Silk road hotel. Still a very nice relaxing option.
Bicycles on the train: This is a more expensive alternative than the bus. But you can sleep at night. Bikes take 1-2 days from Yazd to Teheran for $10,- per bike (If you do this without insurance). Depending of a cargo train going there.
We sent the bike 2 days before. When we arrived in Teheran 4:00 a.m we had to wait untill 8:00 to get the bikes from the cargo department.
We stayed in hotels, camped in the wild and slept in the courtyard of a mosque. everywhere we had a safe feeling.

Answer:
Check the cheapest Iran tours.