The orientational dynamics of inertialess anisotropic particles transported by two-dimensional convective turbulent flows display a coexistence of regular and chaotic features. We numerically demonstrate that very elongated particles (rods) align preferentially with the direction of the fluid flow, i.e., horizontally close to the isothermal walls and dominantly vertically in the bulk. This behaviour is due to the the presence of a persistent large scale circulation flow structure, which induces strong shear at wall boundaries and in up/down-welling regions. The near-wall horizontal alignment of rods persists at increasing the Rayleigh number, while the vertical orientation in the bulk is progressively weakened by the corresponding increase of turbulence intensity. Furthermore, we show that very elongated particles are nearly orthogonal to the orientation of the temperature gradient, an alignment independent of the system dimensionality and which becomes exact only in the limit of infinite Prandtl number. Tumbling rates are extremely vigorous adjacent to the walls, where particles roughly perform Jeffery orbits. This implies that the root-mean-square near-wall tumbling rates for spheres are much stronger than for rods, up to $mathcal{O}(10)$ times at $Rasimeq 10^9$. In the turbulent bulk the situation reverses and rods tumble slightly faster than isotropic particles, in agreement with earlier observations in two-dimensional turbulence.
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