Is transplantation of neurons between species possible?
Short answer
Neural xenografts can survive and form functional connections in the host.

Background
Across-species (xenogeneic grafting) grafting of cerebral cortex from a fetal rat to an adult mouse brain and vice versa, have shown that immunocompetent hosts can reject a graft in less than 30 days. Rejection results in destruction of  the graft tissue presumably by macrophages. Immunosuppression can be used to prevent graft rejection (Mason et al., 1986).

A review article on this topic shows that in the absence of immunosuppression, neural xenografts (donors were fetal to adult-aged) in a rat host brain survive in 37% of the cases in case of a mouse donor, 16% in case of a hamster donor, 9% in case of a rabbit rabbit, 7% from a pig donor, and 0% when human donor tissue was used. Hence, species-separation seems to be a determining factor in xenograft rejection. Induction of MHC-I and MHC-II expression in the xenograft mediates graft rejection in the brain. Hence, despite the fact that the brain is immunopriviliged because of the relative low expression levels of MHC class proteins, it is still sufficient to reject grafts (Pakzaban &amp; Isacson, 1994).

Xenografts have been shown to generate functional afferent and efferent axonal connections with the host brain when grafts of the striatum, substantia nigra, hippocampus and retina were homotopically transplanted. 

Functional (behavioral) studies have confirmed these physiological studies. Rats with Parkinson's disease could be successfully treated with substantia nigra grafting from mouse, hamster, rabbit, pig and even humans. Similarly, retinal xenografts have resulted in functional pupil responses, hippocampal xenografts have restored learning deficits, and suprachiasmatic nucleus xenografts have been shown to restore circadian rhythms (Pakzaban &amp; Isacson, 1994).     

References
- Mason et al., Neurosci (1986); 19(3): 685-94.
- Pakzaban &amp; Isacson, Neurosci (1994); 62(4): 989-1001.