Most conlangs are designed to be at least somewhat naturalistic as they are often inspired by natlangs or intended to be used by concultures. They usually use grammar, phonemes, etc. that could easily exist in natural languages of today. However, there certainly isn't a rule requiring would-be conlangers to limit themselves to established possibilities in natlangs.
Thinking outside the conlanging box is hardly a crime and there is much room for experimentation. Just as artists in music and painting have pushed the envelope with atonality and dadaism while being told that their works were not art by critics, you can buck the trends of conlanging and take it in bold new directions.
Here are a few ideas to ponder:
Imagine a conlang that conflates nouns and verbs into one class. This has actually been attempted many times to varying extents, but it is certainly something rarely seen in natlangs.
Study some linguistic writings on language universals and design your conlang to violate some of them.
Most conlangs (and natlangs) have or are intended to have thousands of root words. Why not go the other way and try to minimize roots, perhaps to 50 or less? This is known as oligosynthesis; it has been attempted a few times but so far not very successfully. (In Wikipedia, see Oligosynthetic language, Natural semantic metalanguage, and Toki Pona.)
Consider new ways to inflect. Japanese inflects its verbs for the amount of deference between the speaker and listener. Perhaps a conlang could inflect verbs for something unusual like the color of the agent.
A slightly more useful idea would be to inflect in relation to the speaker's attitude toward the verbal action. E.g. alistus ((I like the fact that) you exist), alistis ((I hate the fact that) you exist), alistes ((I am indifferent to the fact that) you exist), alistas ((I don't wish to indicate my attitude to the fact that) you exist). This parallels the "pejorative" noun inflections found in natlangs such as Spanish.
Most languages with genders or noun classes will inflect the third-person pronouns for gender; consider doing the same for first and/or second person pronouns.
There is also room for conlangs that use phonemes or phonemic distinctions that are not found in any natural language.