Yes, there is a quest (or series of quests) in each major hold that you perform to becoming a thane.
From Wikia's article on Becoming Thane:
Becoming Thane of Morthal is fairly straightforward. First, you investigate the house that burned down and complete the subsequent quests. The grateful Jarl will tell you that she'd be happy to make you Thane as long as you are "known through her Hold." This translates into helping three different people, all of which can be found in the limited square mileage of Morthal.
From IGN's article on Becoming Thane:
The politics of Markarth are quite complicated, and thus becoming Thane is as well.
One way is to complete the Forsworn Conspiracy by killing Mad King Madanach - the Jarl will thank you for handling him, and give you a quest to retrieve Hrolfdir's Shield. Once you've done that, the Jarl rewards you by allowing you to purchase property. He then adds a followup quest to become Thane of Markarth -- own a house and help 5 citizens of Markarth. After all of this, he will finally make you a Thane, and give you the Blade of the Reach.
Talk to Falk Firebeard in the Blue Palace, and he will give you a quest summoning you to investigate lights and sounds coming out of Wolf Skull Cave which is located in the mountains west of Solitude. After completing the quest, talk to Falk again and he will reward you.
Sometime soon you will receive a letter from a courier asking you to speak to Falk once more. (It is unknown if you can go talk to Falk immediately after and simply force him to give you the next part of the quest).
Falk will give you another quest. Once this quest is done, you may speak directly to the Jarl herself, and she will thank you and give you one more task.
After you have completed this task go back and talk to the Jarl. She will invite you to be a thane of hers but in order to be a Thane, you must first own property in Solitude. This property can be purchased from Falk for 25000 gold pieces. Once you buy the property, go and talk to her, and she will appoint you to be a Thane of Solitude.
This Hold is as straightforward as Morthal: obtain a drink for the Jarl, then kill the assigned Bandit Leader. Complete three quests for locals and you're set. This is the standard formula for the smaller Holds (that is, Holds like Morthal, Windhelm, and Falkreath).
Connected to the Theives Guild Quests.
The objectives you need to complete here are as simple as is Morthal. Obtain the Helm for the Jarl, and then help out three people in his Hold, and he'll see fit to make you Thane.
Connected to the Dark Brotherhood Quests.
Complete all the College of Winterhold quests and you will be made into the archmage of the college.