Are linguistic units organized in conceptual categories?

Yes, there are many patterns of conceptual organization evident in language.It shows up in all sorts of psycholinguistic effects such as priming as well as in the way language structures are related to one another. One such pattern is represented in the WordNet dictionary relating synonyms, hyponyms and hypernyms in what they call synsets. There is also a pattern of conceptual organization evident in grammar - for instance verbs and nouns differentiate between thing-like and action/state-like concepts. Cases often are used in related areas - e.g. in Czech accusative denotes directionality when not used as direct object. Passive constructions are related to active constructions through the concept of agent-patient, etc.

But the key thing here is that we're not dealing with a single hierarchical structure (something like a Linnean taxonomy). Parts of language are related in all sorts of different ways. That's why grammars are so hard to write - you have to choose one pattern of organization. And it's also why it's more efficient to organize a dictionary alphabetically rather than following some sort of conceptual pattern.

Another problem I've hinted at is with what you consider a unit. Construction grammar will think of both 'rules' and 'words' as constructions and consider them as conceptual units whereas more traditional approaches may only think of 'words' as 'units' to which conceptual relationship can be applied.

But there are also many patterns of regularity in language that are clearly not conceptual (or not primarily conceptual). For instance, apart from some subtle symbolism, most phonological relationships are not conceptual. Nor are some things like patterns you only find by analyzing corpora (such as most collocations). Of course, many seeming regularities in language may just be epiphenomenal or their conceptual underpinnings may not present the same way in all speakers.

Finally, you also have to be a bit careful about what you mean by conceptual. In the above, I was drawing on ideas I think were best articulated in Lakoff's 'Women, Fire and Dangerous Things', Langacker's 'Cognitive Grammar' and Talmy's 'Cognitive Semantics'. Conceptual in this context means related to or embedded in our broader cognitive system described outside traditional approaches (Lakoff made that argument most cogently in the first 5-6 chapters of his book - the rest is just filling in the gaps).