there appear to be "no known examples" of undecidable problems that are not proven through "diagonalization" (precisely formalized), or more strongly, apparently diagonalization is the only way of showing uncomputability. in fact diagonalization, formalized, is an even more general property than undecidability, because exactly the same "mechanism" is behind other key mathematical proofs namely Godels theorem and the uncountability of the reals. this is revealed in the Lawvere fixed point formalism/ theorem.
maybe inadvertently, your question is also related or nearly equivalent to the Church-Turing thesis in some subtle or direct ways. the thesis is, roughly, that anything computable is computable by a Turing machine. therefore, equivalently, anything that is not computable is not computable by a Turing machine and all uncomputable Turing languages are reducible to the halting problem and vice versa.
the idea that there exist computations not captured by TMs is considered at best speculative and at worst verging on a "generally not credible" idea by the majority of computer scientists; see eg summary/ "debunking" by Fortnow which also cites several published/ peer reviewed counterclaims. a more recent survey on applicability of Turing machine model and "hypercomputation" etc can be found in 
 Halting problem, uncomputable sets: common mathematical proof? / cstheory.se
 Lawvere, F. William. Diagonal Arguments and Cartesian Closed Categories. Lecture Notes in Mathematics, 92 (1969), 134-145
 UBIQUITY SYMPOSIUM 'WHAT IS COMPUTATION?'
THE ENDURING LEGACY OF THE TURING MACHINE / Fortnow
 Turing's Titanic Machine? / Cooper, communications ACM