Morrowind is one of those old games that cannot be solved by the all-encompassing "follow the green arrow" strategy.
The lack of a quest marker leads to two different situations:
1. You are given clear instructions, like "follow the road to town X, then take the first branch to the right." There's no need for a quest marker since you just have to follow the directions you're given. If you forgot them, you can re-read them in your journal - either by selecting the quest you're on in the quest index, or by retrieving the relevant lines from the topic index.
2. You're given unspecific instructions, like "the object is hidden somewhere in this house", or "he lives in a hut in the Grazelands". In this case, you have to search. There's no green arrow that magically shows you information that your character cannot have. Finding the correct location is a part of the task (and may lead to discovering interesting other things on the way).
For many players, this exploration-oriented gameplay is much more enjoyable than simply following some omniscient green arrow like a dog chasing a torchlight. However, it _can_ of course get frustrating if you simply can't find the location in question, or if the directions were wrong or misleading (this unfortunately happens sometimes in Morrowind). There are two solutions to this: One, use the unofficial patch mod, it takes care of the misleading or wrong directions. Two, visit http://www.uesp.net/wiki/Morrowind:Morrowind, which has all quests and locations listed.