The only problem with north part of Iraq is that it's really hard to buy a bike here.
Since January I live in Erbil, which is capital of Iraqi Kurdistan - autonomous region in the north Iraq (basically everything going north from Mosul/Kirkuk).
Region is safe, there is absolutely no danger to travel inside Kurdistan. A lot of countries (all EU, USA, Canada and few others) have "visa on arrival" which you can get at the airport or on the border. This visa allow you to travel for 15 days across Kurdistan only, you cannot go to other parts of Iraq. It's important to remember about it when you plan to go from Turkey border to Erbil, google maps will always suggest road through Mosul, but you are not allowed to go there (I know people who did it, again soldiers were friendly, but you loose a lot of time on explanation at the checkpoint) and it's not that safe as Kurdistan.
Traveling is very easy. There is a lot of checkpoints, but soldiers are usually lazy and friendly. I traveled quite a lot by car and after few trips you master the "hand wave" and they don't even stop you. Even if they do, they are friendly and very often (especially far from the capital) want you to go out from the car to take a picture ;) I met & hosted a lot of hitchhikers from Europe, bicycle travelers and they all were positive about Kurdistan.
Kurdistan is different than Iraq in every way. There is a lot of nice views, google pictures from Rawanduz, Soran, Shaqlawa, Duhok, Dokan lake and the capital - Erbil. I I'm considering for next year to go home (Poland) buy a bike and come back on it. Then I will spend every weekend for trips.
Concerning time when to come. Now it's too cold I guess, max 5 C during the day and sometimes below 0 during the night. Even worse in the mountains, where is snow. In the summer (July/August) temp reaches 50 C. The best time to visit is during the spring. From middle of March all the way to June. 21st of March is a big Kurdish holiday - Newroz, new year's eve in Kurdish calendar. It's warm enough, so everyone is going out of town to Shaqlawa or Rawanduz for campings, etc. Everything is nice and green. Autumn is ok with temperatures, but everything is burned by the sun so it's closer to desert and air is very dusty.
wow, that's a looong post for a first entry to the forum ;)
Anyway, if you would like to plan trip, I'm sure it's worth it. For now I'm planning to stay littlebit longer in Erbil, so if you need some help, advise with the route or just ask some question about Kurdistan, write me and I will try to help.
If you need a place to stay, there is a lot of people using Couchsurfing so there is a big chance you can find it.
And I found one article about riding here: http://www.motorbikeeurope.com/en/mi...trip-into-irak