Are there temporal logics linear time properties that only have counterexamples that are more complex than a lasso?
Pierre Wolper defined in 1983 extended temporal logic (ETL, in Information and Computation 56, 72–99, doi:10.1016/S0019-9958(83)80051-5), where a temporal operator $\mathcal A(\varphi_1,\dots,\varphi_n)$ can be introduced for a finite-state automaton $\mathcal A$.  The formula is satisfied in an infinite word $u$ at position $i$, i.e. $u,i\models\mathcal A(\varphi_1,\dots,\varphi_n)$, if there exists a finite word $a_{i_1}\cdots a_{i_n}$ in the language of the automaton $\mathcal A$, such that for every $1\leq j\leq n$, $u, i+(j-1)\models\varphi_{i_j}$.  Extended temporal logic has the same expressive power as the linear-time $\mu$-calculus or MSO on infinite words, so it does not answer your question.

One can however go further and allow other languages instead of regular ones in the operator $\mathcal A(\dots)$.  Paul Gastin and Stéphane Demri consider the case of context-free languages at the end of their chapter Specification and Verification using Temporal Logics of Modern applications of automata theory (IISc Research Monographs 2, chapter 15, pages 457–494, World Scientific, 2012, http://www.lsv.ens-cachan.fr/Publis/PAPERS/PDF/DG-iis09.pdf).  You can certainly express non-lasso properties in such a logic.  It is shown by Gastin and Demri to have (highly) undecidable satisfiability and model-checking problems.