  The classical form of Gr\"uss' inequality, first published by G. Gr\"{u}ss in
1935, gives an estimate of the difference between the integral of the product
and the product of the integrals of two functions. In the subsequent years,
many variants of this inequality appeared in the literature. The aim of this
paper is to introduce a different approach, presenting a new
Chebyshev-Gr\"uss-type inequality and applying it to different well-known
linear, not necessarily positive, operators. Some conjectures are presented as
well. We also compare the new inequalities with some older results. This new
approach gives better estimates in some cases than the ones already known.
