  The Lense--Thirring spacetime describes a 4-dimensional slowly rotating
approximate solution of vacuum Einstein equations valid to a linear order in
rotation parameter. It is fully characterized by a single metric function of
the corresponding static (Schwarzschild) solution. In this paper, we introduce
a generalization of the Lense--Thirring spacetimes to the higher-dimensional
multiply-spinning case, with an ansatz that is not necessarily fully
characterized by a single (static) metric function. This generalization lets us
study slowly rotating spacetimes in various higher curvature gravities as well
as in the presence of non-trivial matter. Moreover, the ansatz can be recast in
Painlev{\'e}--Gullstrand form (and thence is manifestly regular on the horizon)
and admits a tower of exact rank-2 and higher rank Killing tensors that rapidly
grows with the number of dimensions. In particular, we construct slowly
multiply-spinning solutions in Lovelock gravity and notably show that in four
dimensions Einstein gravity is the only non-trivial theory amongst all up to
quartic curvature gravities that admits a Lense--Thirring solution
characterized by a single metric function.
