A numerical approach based on the Lattice Boltzmann and Immersed Boundary methods is pro- posed to tackle the problem of the interaction of moving and/or deformable slender solids with an incompressible fluid flow. The method makes use of a Cartesian uniform lattice that encom- passes both the fluid and the solid domains. The deforming/moving elements are tracked through a series of Lagrangian markers that are embedded in the computational domain. Differently from classical projection methods applied to advance in time the incompressible Navier-Stokes equa- tions, the baseline Lattice Boltzmann fluid solver is free from pressure corrector step, which is known to affect the accuracy of the boundary conditions. Also, in contrast to other immersed boundary methods proposed in the literature, the proposed algorithm does not require the in- troduction of any empirical parameter. In the case of rigid bodies, the position of the markers delimiting the surface of an object is updated by tracking both the position of the center of mass of the object and its rotation using Newton's Laws and the conservation of angular momentum. The dynamics of a flexible slender structure is determined as a function of the forces exerted by the fluid, its flexural rigidity and the tension necessary to enforce the filament inextensibility. For both rigid and deformable bodies, the instantaneous no-slip and impermeability conditions on the solid boundary are imposed via external and localized body forces which are consistently introduced into the Lattice Boltzmann equation. The validation test-cases for rigid bodies in- clude the case of an impulsively started plate and the sedimentation of particles under gravity in a fluid initially at rest. For the case of deformable slender structures we consider the beating of both a single filament and a pair filaments induced by the interaction with an incoming uniformly streaming flow.