Algorithmic operator algebras via normal forms for tensors. We propose a general algorithmic approach to noncommutative operator algebras generated by linear operators. Ore algebras are a well-established tool covering many cases arising in applications. However, integro-differential operators, for example, do not fit this structure. Instead of using (parametrized) Gr”obner bases in noncommutative polynomial algebras as has been used so far in the literature, we use {it G. M. Bergman}’s [Adv. Math. 29, 178--218 (1977; Zbl 0326.16019)] basis-free analog in tensor algebras. This allows for a finite reduction system with unique normal forms. To have a smaller reduction system, we develop a generalization of Bergman’s setting, which also makes the algorithmic verification of the confluence criterion more efficient. We provide an implementation in Mathematica and we illustrate both versions of the tensor setting using integro-differential operators as an example.